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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
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l.7     <
         title>MATH2340 Week 4</title>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.13     <
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You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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! You can't use `macro parameter character #' in horizontal mode.
l.17         --uswds-red-cool-20: #
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Sorry, but I'm not programmed to handle this case;
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If you're in the wrong mode, you might be able to
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! You can't use `macro parameter character #' in horizontal mode.
l.18         --uswds-gold-20: #
                               dec69a;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

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Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.19         --uswds-green-cool-20: #
                                     b4d0b9;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no b in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.20         --uswds-blue-cool-20: #
                                    adcfdc;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.21         --uswds-indigo-20: #
                                 c5c5f3;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no c in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.22         --uswds-violet-warm-20: #
                                      e2bee4;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no e in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 6 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.36         background-color: #
                                fff;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no Q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.59             #
                  # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.59             ##
                    Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 13--62
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.63             $
                  f:X\to Y$ has
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <7> on input line 63.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <5> on input line 63.
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (31.52776pt too wide) in paragraph at lines 63--64
[]$\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 63--64
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 63--64
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.65             *
                  domain* $X$, the set of inputs,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 65--66
[]
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 65--66
\OML/cmm/m/it/10 X$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.67             *
                  codomain* $Y$, the set that all the outputs belong to,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 67--68
[]
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 67--68
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.69             *
                  range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ fo...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ) in font nullfont!
! Missing } inserted.
<inserted text> 
                }
l.69 ...     *range* (or *image*) $f(X)=\\{y\in Y$
                                                   such that $y=f(x)$ for so...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
! Extra }, or forgotten $.
<recently read> }
                 
l.69 ... Y$ such that $y=f(x)$ for some $x\in X\\}
                                                  $.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 69--70
[]
 []


Overfull \hbox (33.375pt too wide) in paragraph at lines 69--70
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (25.51149pt too wide) in paragraph at lines 69--70
[]$
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 69--70
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 69--70
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (15.15967pt too wide) in paragraph at lines 69--70
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 69--70
\OML/cmm/m/it/10 X[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.71             C
                  an use a table, a rule, a formula to define a function.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 71--72
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.73             |
                   $x$  | \| | $2$  | $4$   | $6$   | $8$ | $10$ | $12$ |
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no | in font nullfont!
Missing character: There is no | in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.73             | $x$  | \|
                             | $2$  | $4$   | $6$   | $8$ | $10$ | $12$ |
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

Missing character: There is no 2 in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no 6 in font nullfont!
Missing character: There is no 8 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 3 in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.75             | $f(x)$ | \| | $13$ | $\pi
                                            $ | $\frac12$ | $1$ | $1$  | $3$  |
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

Missing character: There is no | in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.78             #
                  # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.78             ##
                    Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 73--81
[]
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 73--81
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (35.0001pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 kj$$j$$j$$j$$j$$j$$jj ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (10.55559pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 j ^^@
 []


Overfull \hbox (10.55559pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (18.33339pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (10.55559pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (10.55559pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (10.55559pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (18.33339pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@j ^^@
 []


Overfull \hbox (15.5556pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@ ^^@
 []


Overfull \hbox (32.7258pt too wide) in paragraph at lines 73--81
\OMS/cmsy/m/n/10 ^^@jj$$jkj$$j$$\OML/cmm/m/it/10 ^^Y$
 []


Overfull \hbox (6.38612pt too wide) in paragraph at lines 73--81
[]$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 73--81
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 73--81
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 73--81
\OT1/cmr/m/n/10 3$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.82             $
                  f:X\to Y$ has
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (31.52776pt too wide) in paragraph at lines 82--83
[]$\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 82--83
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 82--83
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.84             *
                  domain* $X$, the set of inputs,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 84--85
[]
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 84--85
\OML/cmm/m/it/10 X$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.86             *
                  codomain* $Y$, the set that all the outputs belong to,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 86--87
[]
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 86--87
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.88             *
                  range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ fo...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ) in font nullfont!
! Missing } inserted.
<inserted text> 
                }
l.88 ...     *range* (or *image*) $f(X)=\\{y\in Y$
                                                   such that $y=f(x)$ for so...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
! Extra }, or forgotten $.
<recently read> }
                 
l.88 ... Y$ such that $y=f(x)$ for some $x\in X\\}
                                                  $.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 88--89
[]
 []


Overfull \hbox (33.375pt too wide) in paragraph at lines 88--89
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (25.51149pt too wide) in paragraph at lines 88--89
[]$
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 88--89
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 88--89
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (15.15967pt too wide) in paragraph at lines 88--89
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 88--89
\OML/cmm/m/it/10 X[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.90             C
                  an use a table, a rule, a formula to define a function.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 90--91
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.92             $
                  f(x) = x$ if $x$ is an integer, and $f(x) = t$ such that $...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.95             #
                  # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.95             ##
                    Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (50.02084pt too wide) in paragraph at lines 92--98
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (3.61111pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 t$
 []


Overfull \hbox (3.61111pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 t$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 92--98
\OML/cmm/m/it/10 x$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.99             $
                  f:X\to Y$ has
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (31.52776pt too wide) in paragraph at lines 99--100
[]$\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 99--100
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 99--100
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.101             *
                   domain* $X$, the set of inputs,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 101--102
[]
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 101--102
\OML/cmm/m/it/10 X$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.103             *
                   codomain* $Y$, the set that all the outputs belong to,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 103--104
[]
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 103--104
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.105             *
                   range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ f...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ) in font nullfont!
! Missing } inserted.
<inserted text> 
                }
l.105 ...    *range* (or *image*) $f(X)=\\{y\in Y$
                                                   such that $y=f(x)$ for so...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
! Extra }, or forgotten $.
<recently read> }
                 
l.105 ...Y$ such that $y=f(x)$ for some $x\in X\\}
                                                  $.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 105--106
[]
 []


Overfull \hbox (33.375pt too wide) in paragraph at lines 105--106
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (25.51149pt too wide) in paragraph at lines 105--106
[]$
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 105--106
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 105--106
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (15.15967pt too wide) in paragraph at lines 105--106
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 105--106
\OML/cmm/m/it/10 X[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.107             C
                   an use a table, a rule, a formula to define a function.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 107--108
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.109             $
                   f(x) = x^2 - \sin(x)$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.112             #
                   # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.112             ##
                     Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (50.02084pt too wide) in paragraph at lines 109--115
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.9792pt too wide) in paragraph at lines 109--115
\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (25.77087pt too wide) in paragraph at lines 109--115
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.116             $
                   f:X\to Y$ has
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (31.52776pt too wide) in paragraph at lines 116--117
[]$\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 116--117
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 116--117
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.118             *
                   domain* $X$, the set of inputs,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 118--119
[]
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 118--119
\OML/cmm/m/it/10 X$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.120             *
                   codomain* $Y$, the set that all the outputs belong to,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 120--121
[]
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 120--121
\OML/cmm/m/it/10 Y$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.122             *
                   range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ f...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no ) in font nullfont!
! Missing } inserted.
<inserted text> 
                }
l.122 ...    *range* (or *image*) $f(X)=\\{y\in Y$
                                                   such that $y=f(x)$ for so...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
! Extra }, or forgotten $.
<recently read> }
                 
l.122 ...Y$ such that $y=f(x)$ for some $x\in X\\}
                                                  $.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 122--123
[]
 []


Overfull \hbox (33.375pt too wide) in paragraph at lines 122--123
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (25.51149pt too wide) in paragraph at lines 122--123
[]$
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 122--123
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 122--123
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (15.15967pt too wide) in paragraph at lines 122--123
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 122--123
\OML/cmm/m/it/10 X[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.124             C
                   an use a table, a rule, a formula to define a function.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 124--125
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.126             $
                   f(x) = x^2 - \sin(x)$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (50.02084pt too wide) in paragraph at lines 126--127
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.9792pt too wide) in paragraph at lines 126--127
\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (25.77087pt too wide) in paragraph at lines 126--127
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.128             M
                   ost of our functions will be defined by formulae.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no M in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.131             #
                   # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.131             ##
                     Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 128--134
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.135             A
                    function always gives the same output for the same input.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 135--136
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.137             I
                   f $x$ is fixed, then $f(x)$ is fixed.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 137--138
[]
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 137--138
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 137--138
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.139             $
                   f$ is the function, $f(x)$ means "the function $f$ evalua...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.142             #
                   # Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.142             ##
                     Functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (25.97226pt too wide) in paragraph at lines 139--145
[]$\OML/cmm/m/it/10 f$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 139--145
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 139--145
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 139--145
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 139--145
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 139--145
\OML/cmm/m/it/10 x$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.146             A
                    function always gives the same output for the same input.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 146--147
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.148             B
                   ut different inputs could give the same output.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 148--149
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.150             #
                   ### Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.150             ##
                    ## Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.150             ###
                     # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.150             ####
                       Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.150             #### E
                        xample
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.151             $f:\mathbb
                             R\to\mathbb R$ defined by $f(x) = x^2$ has $f(-...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.151             $f:\mathbb R\to\mathbb
                                         R$ defined by $f(x) = x^2$ has $f(-...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 150--152
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 x[]$
 []


Overfull \hbox (37.79865pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 x\OT1/c
mr/m/n/10 ) =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 150--152
\OML/cmm/m/it/10 f$
 []

! You can't use `macro parameter character #' in vertical mode.
l.153             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.153             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.153             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.153             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.153             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.154 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.157             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.157             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 153--160
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 153--160
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 153--160
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 153--160
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 153--160
\OML/cmm/m/it/10 x[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.161             A
                    function always gives the same output for the same input.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 161--162
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.163             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.163             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.163             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.163             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.163             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.164 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 163--165
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 163--165
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 163--165
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 163--165
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 163--165
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.166             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 166--167
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 166--167
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 166--167
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.168             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.171             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.171             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 168--172
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.173             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.173             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.173             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.173             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.173             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.174 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 173--175
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 173--175
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 173--175
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 173--175
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 173--175
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.176             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 176--177
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 176--177
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 176--177
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: The font size command \normalsize is not defined:
               there is probably something wrong with the class file.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.177 
      
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]

! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.178             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.179             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.179             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.179             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.179             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no W in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
<recently read> \mathbb 
                        
l.180 ... surjective, bijective on domain $\mathbb
                                                   R$ and codomain $\mathbb R$?
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
<recently read> \mathbb 
                        
l.180 ... domain $\mathbb R$ and codomain $\mathbb
                                                   R$?
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no ? in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 178--181
[]
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 178--181
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 178--181
\OML/cmm/m/it/10 R$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.182             $
                   f(x) = x^3, \\;\\; g(x) = x^3-x, \\;\\; h(x) = \frac1{\lv...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

! Undefined control sequence.
<argument> \lvert 
                  x \rvert +1
l.182 ... \\;\\; h(x) = \frac1{\lvert x \rvert +1}
                                                  , \\;\\; p(x) = \sin(x)$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert x \rvert 
                           +1
l.182 ... \\;\\; h(x) = \frac1{\lvert x \rvert +1}
                                                  , \\;\\; p(x) = \sin(x)$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!

Overfull \hbox (50.02084pt too wide) in paragraph at lines 182--183
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (14.64581pt too wide) in paragraph at lines 182--183
\OML/cmm/m/it/10 x[]; [] 
 []


Overfull \hbox (4.44441pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; [] 
 []


Overfull \hbox (33.62146pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr
/m/n/10 ) =
 []


Overfull \hbox (17.9792pt too wide) in paragraph at lines 182--183
\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.15968pt too wide) in paragraph at lines 182--183
\OML/cmm/m/it/10 x; [] 
 []


Overfull \hbox (4.44441pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; [] 
 []


Overfull \hbox (34.25458pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 h\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr
/m/n/10 ) =
 []


Overfull \hbox (21.50418pt too wide) in paragraph at lines 182--183
[]\OML/cmm/m/it/10 ; [] 
 []


Overfull \hbox (4.44441pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; [] 
 []


Overfull \hbox (33.52425pt too wide) in paragraph at lines 182--183
\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 p\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr
/m/n/10 ) =
 []


Overfull \hbox (25.77087pt too wide) in paragraph at lines 182--183
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.184             $
                   q(x) = 5, \\;\\; r(x) = x+\lceil x \rceil$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no . in font nullfont!

Overfull \hbox (48.87149pt too wide) in paragraph at lines 184--185
[]$\OML/cmm/m/it/10 q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (9.44443pt too wide) in paragraph at lines 184--185
\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; [] 
 []


Overfull \hbox (4.44441pt too wide) in paragraph at lines 184--185
\OT1/cmr/m/n/10 ; [] 
 []


Overfull \hbox (33.28236pt too wide) in paragraph at lines 184--185
\OT1/cmr/m/n/10 ; \OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr
/m/n/10 ) =
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 184--185
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (14.60416pt too wide) in paragraph at lines 184--185
\OMS/cmsy/m/n/10 d\OML/cmm/m/it/10 x\OMS/cmsy/m/n/10 e$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.186             (
                   How) can you restrict their domains and codomains to make...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ( in font nullfont!
Missing character: There is no H in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ? in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.189             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.189             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 186--190
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.191             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.191             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.191             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.191             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.191             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.192 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 191--193
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 191--193
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 191--193
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 191--193
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 191--193
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.194             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 194--195
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 194--195
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 194--195
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.196             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 196--197
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.198             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 198--200
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.201             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.201             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.201             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.201             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.201             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 201--203
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 201--203
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 201--203
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.204             E
                   very horizontal line crosses the curve $\leq1$ time, so $...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 204--205
[]
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 204--205
\OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 204--205
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 204--205
\OML/cmm/m/it/10 f$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.206             E
                   very horizontal line crosses the curve $\geq1$ time, so $...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 206--207
[]
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 206--207
\OMS/cmsy/m/n/10 ^^U
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 206--207
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 206--207
\OML/cmm/m/it/10 f$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.208             $
                   f$ is both injective and surjective, so bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (25.97226pt too wide) in paragraph at lines 208--211
[]$\OML/cmm/m/it/10 f$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.212             !
                   [Desmos plot of y=x cubed](gfx/x3.png "Desmos plot of y=x...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.212 ...=x cubed](gfx/x3.png "Desmos plot of y=x^
                                                  3")
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.215 
      
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


Overfull \hbox (20.0pt too wide) in paragraph at lines 212--215
[]
 []


Overfull \hbox (23.93056pt too wide) in paragraph at lines 212--215
[]\OT1/cmr/m/n/10 ") \OML/cmm/m/it/10 <
 []


Overfull \hbox (37.18933pt too wide) in paragraph at lines 212--215
\OML/cmm/m/it/10 =div ><
 []


Overfull \hbox (29.41153pt too wide) in paragraph at lines 212--215
\OML/cmm/m/it/10 =div >$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.216             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.218             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.218             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 216--219
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.220             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.220             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.220             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.220             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.220             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.221 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 220--222
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 220--222
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 220--222
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 220--222
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 220--222
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.223             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 223--224
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 223--224
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 223--224
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.225             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 225--226
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.227             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 227--229
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.230             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.230             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.230             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.230             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.230             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 230--232
[]
 []


Overfull \hbox (29.17705pt too wide) in paragraph at lines 230--232
\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.9792pt too wide) in paragraph at lines 230--232
\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 230--232
\OML/cmm/m/it/10 x$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.233             S
                   ome horizontal lines cross the curve $>1$ time, so $g$ is...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 233--234
[]
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 233--234
\OML/cmm/m/it/10 >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 233--234
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.12846pt too wide) in paragraph at lines 233--234
\OML/cmm/m/it/10 g$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.235             E
                   very horizontal line crosses the curve $\geq1$ time, so $...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 235--236
[]
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 235--236
\OMS/cmsy/m/n/10 ^^U
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 235--236
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.12846pt too wide) in paragraph at lines 235--236
\OML/cmm/m/it/10 g$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.237             $
                   g$ is not injective, so it is not bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (25.12846pt too wide) in paragraph at lines 237--240
[]$\OML/cmm/m/it/10 g$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.241             !
                   [Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos p...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.241 ...inus x](gfx/x3-x.png "Desmos plot of y=x^
                                                  3-x")
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.244 
      
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


Overfull \hbox (20.0pt too wide) in paragraph at lines 241--244
[]
 []


Overfull \hbox (12.26393pt too wide) in paragraph at lines 241--244
[] \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (25.1597pt too wide) in paragraph at lines 241--244
\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ") \OML/cmm/m/it/10 <
 []


Overfull \hbox (37.18933pt too wide) in paragraph at lines 241--244
\OML/cmm/m/it/10 =div ><
 []


Overfull \hbox (29.41153pt too wide) in paragraph at lines 241--244
\OML/cmm/m/it/10 =div >$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.245             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.247             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.247             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 245--248
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.249             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.249             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.249             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.249             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.249             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.250 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 249--251
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 249--251
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 249--251
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 249--251
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 249--251
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.252             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 252--253
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 252--253
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 252--253
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.254             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 254--255
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.256             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 256--258
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.259             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.259             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.259             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.259             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.259             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.260             $h:\mathbb
                             R\to\mathbb R$ given by $h(x) = \frac1{\lvert x...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.260             $h:\mathbb R\to\mathbb
                                         R$ given by $h(x) = \frac1{\lvert x...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
<argument> \lvert 
                  x \rvert +1
l.260 ...ven by $h(x) = \frac1{\lvert x \rvert +1}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert x \rvert 
                           +1
l.260 ...ven by $h(x) = \frac1{\lvert x \rvert +1}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 259--263
[]
 []


Overfull \hbox (11.31708pt too wide) in paragraph at lines 259--263
\OML/cmm/m/it/10 h \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 259--263
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 259--263
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (29.81017pt too wide) in paragraph at lines 259--263
\OML/cmm/m/it/10 h\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.05977pt too wide) in paragraph at lines 259--263
[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.264             $
                   h:[0,\infty)\to(0,1]$ would be bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (31.31708pt too wide) in paragraph at lines 264--265
[]$\OML/cmm/m/it/10 h \OT1/cmr/m/n/10 :
 []


Overfull \hbox (38.88885pt too wide) in paragraph at lines 264--265
\OT1/cmr/m/n/10 [0\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 ) \OMS/c
msy/m/n/10 !
 []


Overfull \hbox (21.11113pt too wide) in paragraph at lines 264--265
\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.266             $
                   h:(-\infty,0]\to(0,1]$ would be bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (31.31708pt too wide) in paragraph at lines 266--269
[]$\OML/cmm/m/it/10 h \OT1/cmr/m/n/10 :
 []


Overfull \hbox (46.66666pt too wide) in paragraph at lines 266--269
\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0] \OM
S/cmsy/m/n/10 !
 []


Overfull \hbox (21.11113pt too wide) in paragraph at lines 266--269
\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.270             !
                   [Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.p...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 270--273
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.274             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.276             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.276             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 274--277
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.278             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.278             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.278             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.278             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.278             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.279 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 278--280
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 278--280
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 278--280
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 278--280
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 278--280
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.281             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 281--282
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 281--282
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 281--282
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.283             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 283--284
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.285             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 285--287
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.288             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.288             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.288             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.288             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.288             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.289             $p:\mathbb
                             R\to\mathbb R$ given by $p(x) = \sin(x)$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.289             $p:\mathbb R\to\mathbb
                                         R$ given by $p(x) = \sin(x)$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 288--292
[]
 []


Overfull \hbox (10.58675pt too wide) in paragraph at lines 288--292
\OML/cmm/m/it/10 p \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 288--292
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 288--292
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (29.07983pt too wide) in paragraph at lines 288--292
\OML/cmm/m/it/10 p\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (25.77087pt too wide) in paragraph at lines 288--292
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.293             $
                   p:[\frac{-\pi}2,\frac\pi2]\to[-1,1]$ would be bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (30.58675pt too wide) in paragraph at lines 293--294
[]$\OML/cmm/m/it/10 p \OT1/cmr/m/n/10 :
 []


Overfull \hbox (43.68266pt too wide) in paragraph at lines 293--294
\OT1/cmr/m/n/10 [[]\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/10 ] \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (27.77782pt too wide) in paragraph at lines 293--294
\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/c
mr/m/n/10 1]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.295             $
                   p:[\frac{-3\pi}2,\frac{-\pi}2]\to[-1,1]$ would be bijective.

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (30.58675pt too wide) in paragraph at lines 295--296
[]$\OML/cmm/m/it/10 p \OT1/cmr/m/n/10 :
 []


Overfull \hbox (53.91881pt too wide) in paragraph at lines 295--296
\OT1/cmr/m/n/10 [[]\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/10 ] \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (27.77782pt too wide) in paragraph at lines 295--296
\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/c
mr/m/n/10 1]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.297             $
                   p:[-\pi,0]\to[-1,0]$ would be surjective but not injective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (30.58675pt too wide) in paragraph at lines 297--300
[]$\OML/cmm/m/it/10 p \OT1/cmr/m/n/10 :
 []


Overfull \hbox (41.61458pt too wide) in paragraph at lines 297--300
\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 ^^Y; \OT1/cmr/m/n/10 0] \
OMS/cmsy/m/n/10 !
 []


Overfull \hbox (27.77782pt too wide) in paragraph at lines 297--300
\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/c
mr/m/n/10 0]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.301             !
                   [Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 301--304
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.305             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.307             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.307             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 305--308
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.309             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.309             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.309             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.309             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.309             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.310 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 309--311
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 309--311
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 309--311
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 309--311
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 309--311
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.312             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 312--313
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 312--313
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 312--313
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.314             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 314--315
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.316             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 316--318
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.319             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.319             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.319             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.319             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.319             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.320             $q:\mathbb
                             R\to\mathbb R$ given by $q(x) = 5$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.320             $q:\mathbb R\to\mathbb
                                         R$ given by $q(x) = 5$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 319--322
[]
 []


Overfull \hbox (10.3784pt too wide) in paragraph at lines 319--322
\OML/cmm/m/it/10 q \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 319--322
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 319--322
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (28.87149pt too wide) in paragraph at lines 319--322
\OML/cmm/m/it/10 q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 319--322
\OT1/cmr/m/n/10 5$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.323             T
                   o make $q$ bijective, we would have to its domain and cod...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 323--326
[]
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 323--326
\OML/cmm/m/it/10 q$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.328             <
                   /div>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 328--330
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.331             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.333             #
                   # Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.333             ##
                     Functions: injectivity and surjectivity
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no F in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 331--334
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.335             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.335             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.335             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.335             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.335             #### D
                        efinition
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.336 ...is *injective* if $f(x_1)=f(x_2) \implies
                                                   x_1=x_2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 335--337
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 335--337
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (34.50697pt too wide) in paragraph at lines 335--337
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (44.70837pt too wide) in paragraph at lines 335--337
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cm
m/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 335--337
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.338             A
                    function $f:X\to Y$ is *surjective* if its range is its ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 338--339
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (21.84715pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (8.02779pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 Y$
 []


Overfull \hbox (24.70598pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8\OML/cmm/m/it/10 y \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (10.80553pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 Y; [] 
 []


Overfull \hbox (9.99998pt too wide) in paragraph at lines 338--339
\OT1/cmr/m/n/10 ; \OMS/cmsy/m/n/10 9[] 
 []


Overfull \hbox (19.60408pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 ; x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (9.06943pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 X$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.2616pt too wide) in paragraph at lines 338--339
\OML/cmm/m/it/10 y$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.340             A
                    function is *bijective* if it is both injective and surj...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 340--341
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.342             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 342--344
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.345             #
                   ### Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.345             ##
                    ## Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.345             ###
                     # Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.345             ####
                       Examples
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.345             #### E
                        xamples
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.346             $r:\mathbb
                             R\to\mathbb R$ given by $r(x) = x+\lceil x\rceil$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.346             $r:\mathbb R\to\mathbb
                                         R$ given by $r(x) = x+\lceil x\rceil$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 345--349
[]
 []


Overfull \hbox (10.34486pt too wide) in paragraph at lines 345--349
\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 345--349
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 345--349
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (28.83795pt too wide) in paragraph at lines 345--349
\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 345--349
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (14.60416pt too wide) in paragraph at lines 345--349
\OMS/cmsy/m/n/10 d\OML/cmm/m/it/10 x\OMS/cmsy/m/n/10 e$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.350             $
                   r:(0,1]\to(1,2]$ would be bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (30.34486pt too wide) in paragraph at lines 350--351
[]$\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 :
 []


Overfull \hbox (33.88885pt too wide) in paragraph at lines 350--351
\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1] \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (21.11113pt too wide) in paragraph at lines 350--351
\OT1/cmr/m/n/10 (1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.352             $
                   r:(0,2]\to(1,2]\cup(3,4]$ is bijective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (30.34486pt too wide) in paragraph at lines 352--353
[]$\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 :
 []


Overfull \hbox (33.88885pt too wide) in paragraph at lines 352--353
\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2] \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (27.77782pt too wide) in paragraph at lines 352--353
\OT1/cmr/m/n/10 (1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2] \OMS/cmsy/m/n/10 [
 []


Overfull \hbox (21.11113pt too wide) in paragraph at lines 352--353
\OT1/cmr/m/n/10 (3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 4]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.354             $
                   r:(0,2]\to(1,4]$ is injective but not surjective.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (30.34486pt too wide) in paragraph at lines 354--357
[]$\OML/cmm/m/it/10 r \OT1/cmr/m/n/10 :
 []


Overfull \hbox (33.88885pt too wide) in paragraph at lines 354--357
\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2] \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (21.11113pt too wide) in paragraph at lines 354--357
\OT1/cmr/m/n/10 (1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 4]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.358             !
                   [Desmos plot of y=x plus the ceiling function of x](gfx/x...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 358--361
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.362             <
                   !-- .slide: class="smallertext" data-auto-animate -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.364             #
                   # Limit estimation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.364             ##
                     Limit estimation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
l.365             Let $f:\mathbb
                                 R \to\mathbb R$ and $a\in\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.365             Let $f:\mathbb R \to\mathbb
                                              R$ and $a\in\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.365 ...:\mathbb R \to\mathbb R$ and $a\in\mathbb
                                                   R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 362--366
[]
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 362--366
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 362--366
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 362--366
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (14.73029pt too wide) in paragraph at lines 362--366
\OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 362--366
\OML/cmm/m/it/10 R$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.367             I
                   nformally, $\displaystyle \ell=\lim_{x\to a}f(x)$ is "the...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 367--368
[]
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 367--368
\OML/cmm/m/it/10 ` \OT1/cmr/m/n/10 =
 []


Overfull \hbox (37.97658pt too wide) in paragraph at lines 367--368
[] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 367--368
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 367--368
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 367--368
\OML/cmm/m/it/10 a$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.369             W
                   e need a more formal definition of limits for this course...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no W in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 369--370
[]
 []


Overfull \hbox (4.16669pt too wide) in paragraph at lines 369--370
\OML/cmm/m/it/10 `$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.371             W
                   e need to prove what $\ell$ is because estimating can res...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no W in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 371--372
[]
 []


Overfull \hbox (4.16669pt too wide) in paragraph at lines 371--372
\OML/cmm/m/it/10 `$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.373             [
                   Excel file demonstrating the dangers of estimating limits...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no [ in font nullfont!
Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no H in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no H in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no Q in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no W in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no J in font nullfont!
Missing character: There is no B in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.376             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.376             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.377             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.377             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.377             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.377             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.378             Suppose $c\lt
                                a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.378             Suppose $c\lt a\lt
                                     d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.378 ... a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb
                                                   R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 373--380
[]
 []


Overfull \hbox (14.81831pt too wide) in paragraph at lines 373--380
\OML/cmm/m/it/10 cad$
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 373--380
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (28.50235pt too wide) in paragraph at lines 373--380
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; a\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 [
 []


Overfull \hbox (38.26848pt too wide) in paragraph at lines 373--380
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a; d\OT1/cmr/m/n/10 ) :\OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 373--380
\OML/cmm/m/it/10 R$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.381             $
                   f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.381 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (39.46533pt too wide) in paragraph at lines 381--382
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


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\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 381--382
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 381--382
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 381--382
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.383             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.383 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.383 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.383 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.383 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.383 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.383 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.383 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.383 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.383 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 383--384
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 383--384
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 383--384
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 383--384
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 383--384
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 383--384
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 383--384
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 383--384
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
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Overfull \hbox (14.7222pt too wide) in paragraph at lines 383--384
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 383--384
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.385             #
                   ### Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.385             ##
                    ## Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.385             ###
                     # Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.385             ####
                       Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.385             #### O
                        r, in words,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no O in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
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Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
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Missing character: There is no o in font nullfont!
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Missing character: There is no - in font nullfont!
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Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no N in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 385--387
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 385--387
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 385--387
\OT1/cmr/m/n/10 0$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.388             <
                   span style="color:var(--uswds-indigo-20)">we can always g...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no n in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no a in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
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Missing character: There is no o in font nullfont!
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Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ) in font nullfont!
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Missing character: There is no o in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
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Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 388--389
[]
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 388--389
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 388--389
\OML/cmm/m/it/10 ^^N$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.390             <
                   span style="color:var(--uswds-green-cool-20)">all $x$ clo...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no a in font nullfont!
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Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
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Missing character: There is no ) in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
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Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no l in font nullfont!
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Missing character: There is no 0 in font nullfont!
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Missing character: There is no " in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 390--392
[]
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 "$
 []


Overfull \hbox (4.16669pt too wide) in paragraph at lines 390--392
\OML/cmm/m/it/10 `$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.393             #
                   ### Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.393             ##
                    ## Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.393             ###
                     # Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.393             ####
                       Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.393             #### N
                        otation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 393--395
[]
 []


Overfull \hbox (72.133pt too wide) in paragraph at lines 393--395
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (4.16669pt too wide) in paragraph at lines 393--395
\OML/cmm/m/it/10 `$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.396             [
                   Excel file demonstrating how a limit is defined](https://...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no [ in font nullfont!
Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no Y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no E in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no C in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.399             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.399             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.400             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.400             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.400             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.400             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.401             Suppose $c\lt
                                a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.401             Suppose $c\lt a\lt
                                     d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.401 ... a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb
                                                   R$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 396--403
[]
 []


Overfull \hbox (14.81831pt too wide) in paragraph at lines 396--403
\OML/cmm/m/it/10 cad$
 []


Overfull \hbox (11.52776pt too wide) in paragraph at lines 396--403
\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 :
 []


Overfull \hbox (28.50235pt too wide) in paragraph at lines 396--403
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c; a\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 [
 []


Overfull \hbox (38.26848pt too wide) in paragraph at lines 396--403
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a; d\OT1/cmr/m/n/10 ) :\OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 396--403
\OML/cmm/m/it/10 R$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.404             $
                   f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.404 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (39.46533pt too wide) in paragraph at lines 404--405
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 404--405
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 404--405
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 404--405
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 404--405
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.406             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.406 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.406 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.406 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.406 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.406 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.406 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.406 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.406 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.406 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 406--407
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 406--407
\OML/cmm/m/it/10 " >
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Overfull \hbox (5.00002pt too wide) in paragraph at lines 406--407
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 406--407
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 406--407
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 406--407
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 406--407
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 406--407
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 406--407
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 406--407
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.408             #
                   ### Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.408             ##
                    ## Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.408             ###
                     # Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.408             ####
                       Or, in words,
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.408             #### O
                        r, in words,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no O in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 408--410
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 408--410
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 408--410
\OT1/cmr/m/n/10 0$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.411             <
                   span style="color:var(--uswds-indigo-20)">we can always g...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 411--412
[]
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 411--412
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 411--412
\OML/cmm/m/it/10 ^^N$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.413             <
                   span style="color:var(--uswds-green-cool-20)">all $x$ clo...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.415             #
                   ### How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.415             ##
                    ## How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.415             ###
                     # How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.415             ####
                       How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no H in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.419             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.419             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.420             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.420             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.420             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.420             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.421 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 413--422
[]
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\OML/cmm/m/it/10 x$
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\OML/cmm/m/it/10 a$
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\OML/cmm/m/it/10 ^^N$
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\OML/cmm/m/it/10 a$
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Overfull \hbox (19.46533pt too wide) in paragraph at lines 413--422
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


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\OML/cmm/m/it/10 "$
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\OML/cmm/m/it/10 `$
 []


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\OML/cmm/m/it/10 ^^N$
 []


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\OML/cmm/m/it/10 "$
 []


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\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 413--422
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


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\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
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\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 413--422
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 413--422
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.423             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.423 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.423 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.423 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.423 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.423 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.423 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.423 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.423 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.423 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 423--424
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 423--424
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 423--424
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 423--424
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 423--424
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 423--424
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 423--424
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 423--424
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


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\OML/cmm/m/it/10 ` <
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\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.425             #
                   ### How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.425             ##
                    ## How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.425             ###
                     # How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.425             ####
                       How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.425             #### H
                        ow to prove convergence if you know the limit
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no H in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 425--427
[]
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 425--427
\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 425--427
\OML/cmm/m/it/10 "$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 425--427
\OML/cmm/m/it/10 ^^N$
 []

! You can't use `macro parameter character #' in vertical mode.
l.428             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.428             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.428             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.428             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.428             #### T
                        heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.430             #
                   ### Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.430             ##
                    ## Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.430             ###
                     # Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.430             ####
                       Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.432             If $\lvert
                             x-0 \rvert \lt \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.432             If $\lvert x-0 \rvert
                                        \lt \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.432             If $\lvert x-0 \rvert \lt
                                            \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.433             $\left\lvert
                               f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.433             $\left\lvert f
                                (x)-0 \right\rvert = \lvert x^2 \rvert = x^2...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.433             $\left\lvert f(x)-0 \right\rvert
                                                   = \lvert x^2 \rvert = x^2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.433 ...       $\left\lvert f(x)-0 \right\rvert =
                                                   \lvert x^2 \rvert = x^2 \...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.433 ...$\left\lvert f(x)-0 \right\rvert = \lvert
                                                   x^2 \rvert = x^2 \lt \del...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.433 ...t f(x)-0 \right\rvert = \lvert x^2 \rvert
                                                   = x^2 \lt \delta^2 = \var...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.433 ...right\rvert = \lvert x^2 \rvert = x^2 \lt
                                                   \delta^2 = \varepsilon$. ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
! Misplaced alignment tab character &.
l.433 ...vert = x^2 \lt \delta^2 = \varepsilon$. &
                                                  nbsp; □
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!

! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.433 ...2 \lt \delta^2 = \varepsilon$. &nbsp; □
                                                  
You may provide a definition with
\DeclareUnicodeCharacter 

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.436             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.436             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.437             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.437             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.437             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.437             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.438 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 428--439
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 x[]$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 428--439
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 428--439
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 " >
 []


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\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.99654pt too wide) in paragraph at lines 428--439
[]$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


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\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ^^N$
 []


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[] \OT1/cmr/m/n/10 =
 []


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\OML/cmm/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


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\OML/cmm/m/it/10 x[]^^N[] \OT1/cmr/m/n/10 =
 []


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\OML/cmm/m/it/10 "$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 428--439
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


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\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


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\OML/cmm/m/it/10 R$
 []


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\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


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\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.440             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.440 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.440 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.440 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.440 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.440 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.440 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.440 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.440 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.440 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 440--441
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 440--441
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 440--441
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 440--441
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 440--441
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 440--441
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 440--441
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 440--441
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 440--441
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 440--441
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.442             #
                   ### How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.442             ##
                    ## How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.442             ###
                     # How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.442             ####
                       How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.442             #### H
                        ow to prove convergence if you know the limit
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no H in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

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[]
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\OML/cmm/m/it/10 ^^N$
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\OML/cmm/m/it/10 "$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 442--444
\OML/cmm/m/it/10 ^^N$
 []

! You can't use `macro parameter character #' in vertical mode.
l.445             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.445             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.445             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.445             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.445             #### T
                        heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.447             #
                   ### Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.447             ##
                    ## Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.447             ###
                     # Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.447             ####
                       Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no R in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.449             $\left\lvert
                               f(x)-0 \right\rvert \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.449             $\left\lvert f
                                (x)-0 \right\rvert \lt \varepsilon$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.449             $\left\lvert f(x)-0 \right\rvert
                                                   \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.449 ...     $\left\lvert f(x)-0 \right\rvert \lt
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.449 ...lvert f(x)-0 \right\rvert \lt \varepsilon
                                                  $.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.450             I calculate $\left\lvert
                                           f(x)-0 \right\rvert = \left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.450             I calculate $\left\lvert f
                                            (x)-0 \right\rvert = \left\lvert...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.450 ...alculate $\left\lvert f(x)-0 \right\rvert
                                                   = \left\lvert x^2 \right\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.450 ...culate $\left\lvert f(x)-0 \right\rvert =
                                                   \left\lvert x^2 \right\rv...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.450 ...t\lvert f(x)-0 \right\rvert = \left\lvert
                                                   x^2 \right\rvert = x^2$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   x
l.450 ...lvert f(x)-0 \right\rvert = \left\lvert x
                                                  ^2 \right\rvert = x^2$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.450 ...ight\rvert = \left\lvert x^2 \right\rvert
                                                   = x^2$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.450 ...ht\rvert = \left\lvert x^2 \right\rvert =
                                                   x^2$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


Overfull \hbox (20.0pt too wide) in paragraph at lines 445--451
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 445--451
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


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\OML/cmm/m/it/10 x[]$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 445--451
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 445--451
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (45.41728pt too wide) in paragraph at lines 445--451
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (49.64299pt too wide) in paragraph at lines 445--451
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (23.1569pt too wide) in paragraph at lines 445--451
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 445--451
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.452             a
                   nd I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.452 ...        and I'm allowed to insist $\lvert
                                                   x-0 \rvert \lt \delta$, w...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.452 ... I'm allowed to insist $\lvert x-0 \rvert
                                                   \lt \delta$, which implie...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.452 ... allowed to insist $\lvert x-0 \rvert \lt
                                                   \delta$, which implies $x...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.452 ...rvert \lt \delta$, which implies $x^2 \lt
                                                   \delta^2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 452--453
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 452--453
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 452--453
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (19.51044pt too wide) in paragraph at lines 452--453
\OML/cmm/m/it/10 x[]^^N[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.454             S
                   o I just need $\delta^2 \leqslant \varepsilon$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.454 ...       So I just need $\delta^2 \leqslant
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.458             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.458             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.459             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.459             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.459             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.459             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.460 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 454--461
[]
 []


Overfull \hbox (13.97221pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 ^^N[]"$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.99654pt too wide) in paragraph at lines 454--461
[]$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


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\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 454--461
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.462             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.462 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.462 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.462 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.462 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.462 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.462 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.462 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.462 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.462 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

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[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 462--463
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 462--463
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 462--463
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 462--463
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 462--463
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 462--463
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 462--463
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 462--463
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 462--463
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.464             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.464             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.464             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.464             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.464             #### T
                        heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.466             #
                   ### Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.466             ##
                    ## Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.466             ###
                     # Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.466             ####
                       Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no R in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.468             $\left\lvert
                               f(x)-0 \right\rvert \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.468             $\left\lvert f
                                (x)-0 \right\rvert \lt \varepsilon$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.468             $\left\lvert f(x)-0 \right\rvert
                                                   \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.468 ...     $\left\lvert f(x)-0 \right\rvert \lt
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.468 ...lvert f(x)-0 \right\rvert \lt \varepsilon
                                                  $.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.469             I calculate $\left\lvert
                                           f(x)-0 \right\rvert = \left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.469             I calculate $\left\lvert f
                                            (x)-0 \right\rvert = \left\lvert...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.469 ...alculate $\left\lvert f(x)-0 \right\rvert
                                                   = \left\lvert x^2 \right\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.469 ...culate $\left\lvert f(x)-0 \right\rvert =
                                                   \left\lvert x^2 \right\rv...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.469 ...t\lvert f(x)-0 \right\rvert = \left\lvert
                                                   x^2 \right\rvert = x^2$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   x
l.469 ...lvert f(x)-0 \right\rvert = \left\lvert x
                                                  ^2 \right\rvert = x^2$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.469 ...ight\rvert = \left\lvert x^2 \right\rvert
                                                   = x^2$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.469 ...ht\rvert = \left\lvert x^2 \right\rvert =
                                                   x^2$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


Overfull \hbox (20.0pt too wide) in paragraph at lines 464--470
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 464--470
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 464--470
\OML/cmm/m/it/10 x[]$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 464--470
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 464--470
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (45.41728pt too wide) in paragraph at lines 464--470
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (49.64299pt too wide) in paragraph at lines 464--470
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (23.1569pt too wide) in paragraph at lines 464--470
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (10.2014pt too wide) in paragraph at lines 464--470
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.471             a
                   nd I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.471 ...        and I'm allowed to insist $\lvert
                                                   x-0 \rvert \lt \delta$, w...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.471 ... I'm allowed to insist $\lvert x-0 \rvert
                                                   \lt \delta$, which implie...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.471 ... allowed to insist $\lvert x-0 \rvert \lt
                                                   \delta$, which implies $x...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.471 ...rvert \lt \delta$, which implies $x^2 \lt
                                                   \delta^2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 471--472
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 471--472
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 471--472
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (19.51044pt too wide) in paragraph at lines 471--472
\OML/cmm/m/it/10 x[]^^N[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.473             S
                   o I just need $\delta^2 \leqslant \varepsilon$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.473 ...       So I just need $\delta^2 \leqslant
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.475             #
                   ### Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.475             ##
                    ## Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.475             ###
                     # Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.475             ####
                       Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.477             If $\lvert
                             x-0 \rvert \lt \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.477             If $\lvert x-0 \rvert
                                        \lt \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.477             If $\lvert x-0 \rvert \lt
                                            \delta$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.478             $\left\lvert
                               f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.478             $\left\lvert f
                                (x)-0 \right\rvert = \lvert x^2 \rvert = x^2...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.478             $\left\lvert f(x)-0 \right\rvert
                                                   = \lvert x^2 \rvert = x^2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.478 ...       $\left\lvert f(x)-0 \right\rvert =
                                                   \lvert x^2 \rvert = x^2 \...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.478 ...$\left\lvert f(x)-0 \right\rvert = \lvert
                                                   x^2 \rvert = x^2 \lt \del...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.478 ...t f(x)-0 \right\rvert = \lvert x^2 \rvert
                                                   = x^2 \lt \delta^2 = \var...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.478 ...right\rvert = \lvert x^2 \rvert = x^2 \lt
                                                   \delta^2 = \varepsilon$. ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
! Misplaced alignment tab character &.
l.478 ...vert = x^2 \lt \delta^2 = \varepsilon$. &
                                                  nbsp; □
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!

! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.478 ...2 \lt \delta^2 = \varepsilon$. &nbsp; □
                                                  
You may provide a definition with
\DeclareUnicodeCharacter 

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.481             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.481             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.482             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.482             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.482             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.482             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.483 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 473--484
[]
 []


Overfull \hbox (13.97221pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 ^^N[]"$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.99654pt too wide) in paragraph at lines 473--484
[]$
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 473--484
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.99654pt too wide) in paragraph at lines 473--484
[]$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 473--484
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (49.64299pt too wide) in paragraph at lines 473--484
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (20.75691pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 x[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (30.06595pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 x[]^^N[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 "$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 473--484
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.485             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.485 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.485 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.485 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.485 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.485 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.485 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.485 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.485 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.485 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 485--486
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 485--486
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 485--486
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 485--486
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 485--486
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 485--486
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 485--486
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 485--486
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 485--486
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 485--486
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.487             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.487             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.487             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.487             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.487             #### T
                        heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.489             #
                   ### Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.489             ##
                    ## Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.489             ###
                     # Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.489             ####
                       Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 487--491
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 487--491
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.17087pt too wide) in paragraph at lines 487--491
[]$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 487--491
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 487--491
\OT1/cmr/m/n/10 2$
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 487--491
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 487--491
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 487--491
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (18.33334pt too wide) in paragraph at lines 487--491
[] [] 
 []


Overfull \hbox (17.21666pt too wide) in paragraph at lines 487--491
[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.492             I
                   f $\lvert x-1 \rvert \lt \delta$, then $6\lvert x-1 \rver...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.492             If $\lvert
                             x-1 \rvert \lt \delta$, then $6\lvert x-1 \rver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492             If $\lvert x-1 \rvert
                                        \lt \delta$, then $6\lvert x-1 \rver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492             If $\lvert x-1 \rvert \lt
                                            \delta$, then $6\lvert x-1 \rver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.492 ...ert x-1 \rvert \lt \delta$, then $6\lvert
                                                   x-1 \rvert\lt\varepsilon$...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492 ...ert \lt \delta$, then $6\lvert x-1 \rvert
                                                  \lt\varepsilon$ and $\lver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492 ... \lt \delta$, then $6\lvert x-1 \rvert\lt
                                                  \varepsilon$ and $\lvert 2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.492 ...ert x-1 \rvert\lt\varepsilon$ and $\lvert
                                                   2-x \rvert = 2-x \gt \fra...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492 ...ert\lt\varepsilon$ and $\lvert 2-x \rvert
                                                   = 2-x \gt \frac12$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.492 ...epsilon$ and $\lvert 2-x \rvert = 2-x \gt
                                                   \frac12$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 492--493
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 492--493
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 492--493
\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 492--493
\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.6632pt too wide) in paragraph at lines 492--493
\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 "$
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 492--493
\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (16.27078pt too wide) in paragraph at lines 492--493
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 492--493
\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (12.1014pt too wide) in paragraph at lines 492--493
\OML/cmm/m/it/10 x[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.494             s
                   o
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
! Undefined control sequence.
l.495             $\left\lvert
                              \frac{x+1}{2-x} - 2 \right\rvert = \frac{3\lve...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   {
l.495             $\left\lvert\frac{x+1}{2-x}
                                              - 2 \right\rvert = \frac{3\lve...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.495 ...eft\lvert\frac{x+1}{2-x} - 2 \right\rvert
                                                   = \frac{3\lvert x-1 \rver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.495 ...t\lvert\frac{x+1}{2-x} - 2 \right\rvert =
                                                   \frac{3\lvert x-1 \rvert}...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> 3\lvert 
                   x-1 \rvert 
l.495 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt \frac{3\lvert x-1 \rv...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> 3\lvert x-1 \rvert 
                              
l.495 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt \frac{3\lvert x-1 \rv...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 
                  2-x \rvert 
l.495 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt \frac{3\lvert x-1 \rv...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 2-x \rvert 
                             
l.495 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt \frac{3\lvert x-1 \rv...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.495 ...\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt
                                                   \frac{3\lvert x-1 \rvert}...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> 3\lvert 
                   x-1 \rvert 
l.495 ...t} \lt \frac{3\lvert x-1 \rvert}{\frac12}
                                                   = 6 \lvert x-1 \rvert\lt\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> 3\lvert x-1 \rvert 
                              
l.495 ...t} \lt \frac{3\lvert x-1 \rvert}{\frac12}
                                                   = 6 \lvert x-1 \rvert\lt\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.495 ...c{3\lvert x-1 \rvert}{\frac12} = 6 \lvert
                                                   x-1 \rvert\lt\varepsilon$...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.495 ...-1 \rvert}{\frac12} = 6 \lvert x-1 \rvert
                                                  \lt\varepsilon$. &nbsp; □
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.495 ...\rvert}{\frac12} = 6 \lvert x-1 \rvert\lt
                                                  \varepsilon$. &nbsp; □
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
! Misplaced alignment tab character &.
l.495 ...} = 6 \lvert x-1 \rvert\lt\varepsilon$. &
                                                  nbsp; □
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!

! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.495 ...ert x-1 \rvert\lt\varepsilon$. &nbsp; □
                                                  
You may provide a definition with
\DeclareUnicodeCharacter 

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.498             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.498             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.499             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.499             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.499             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.499             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.501             #
                   ### Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.501             ##
                    ## Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.501             ###
                     # Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.501             ####
                       Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no R in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.503             $\left\lvert
                               f(x)-2 \right\rvert \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.503             $\left\lvert f
                                (x)-2 \right\rvert \lt \varepsilon$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.503             $\left\lvert f(x)-2 \right\rvert
                                                   \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.503 ...     $\left\lvert f(x)-2 \right\rvert \lt
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.503 ...lvert f(x)-2 \right\rvert \lt \varepsilon
                                                  $.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.504             I calculate $\left\lvert
                                           f(x)-2 \right\rvert = \ldots = \f...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.504             I calculate $\left\lvert f
                                            (x)-2 \right\rvert = \ldots = \f...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.504 ...alculate $\left\lvert f(x)-2 \right\rvert
                                                   = \ldots = \frac{3\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.504 ...culate $\left\lvert f(x)-2 \right\rvert =
                                                   \ldots = \frac{3\lvert x-...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> 3\lvert 
                   x-1 \rvert 
l.504 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> 3\lvert x-1 \rvert 
                              
l.504 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 
                  2-x \rvert 
l.504 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 2-x \rvert 
                             
l.504 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


Overfull \hbox (20.0pt too wide) in paragraph at lines 494--505
[]
 []


Overfull \hbox (47.34853pt too wide) in paragraph at lines 494--505
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (52.8695pt too wide) in paragraph at lines 494--505
[][] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 494--505
\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.6632pt too wide) in paragraph at lines 494--505
\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 "$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 494--505
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (17.17087pt too wide) in paragraph at lines 494--505
[]$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 494--505
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 494--505
\OT1/cmr/m/n/10 2$
 []


Overfull \hbox (45.41728pt too wide) in paragraph at lines 494--505
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (49.64299pt too wide) in paragraph at lines 494--505
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (22.22212pt too wide) in paragraph at lines 494--505
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (21.157pt too wide) in paragraph at lines 494--505
[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.506             a
                   nd I'm allowed to insist $\lvert x-1 \rvert \lt \delta$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.506 ...        and I'm allowed to insist $\lvert
                                                   x-1 \rvert \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.506 ... I'm allowed to insist $\lvert x-1 \rvert
                                                   \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.506 ... allowed to insist $\lvert x-1 \rvert \lt
                                                   \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.507 ...      If $x$ is close to $1$ then $\lvert
                                                   2-x \rvert$ is close to $...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.507 ...$ is close to $1$ then $\lvert 2-x \rvert
                                                  $ is close to $1$ too.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 506--508
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 506--508
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 506--508
\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 506--508
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 506--508
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 506--508
\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (5.71527pt too wide) in paragraph at lines 506--508
\OML/cmm/m/it/10 x$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 506--508
\OT1/cmr/m/n/10 1$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.509             Q
                   uantitatively, if $\lvert x-1 \rvert \lt \frac12$, then $...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no Q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.509             Quantitatively, if $\lvert
                                             x-1 \rvert \lt \frac12$, then $...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ...    Quantitatively, if $\lvert x-1 \rvert
                                                   \lt \frac12$, then $\frac...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ...Quantitatively, if $\lvert x-1 \rvert \lt
                                                   \frac12$, then $\frac12 \...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.509 ...-1 \rvert \lt \frac12$, then $\frac12 \lt
                                                   \lvert 2-x \rvert \lt \fr...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ...rt \lt \frac12$, then $\frac12 \lt \lvert
                                                   2-x \rvert \lt \frac32$, ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ...c12$, then $\frac12 \lt \lvert 2-x \rvert
                                                   \lt \frac32$, so $\frac1{...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ..., then $\frac12 \lt \lvert 2-x \rvert \lt
                                                   \frac32$, so $\frac1{\lve...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
! Undefined control sequence.
<argument> \lvert 
                  2-x \rvert 
l.509 ...t \frac32$, so $\frac1{\lvert 2-x \rvert}
                                                   \lt 2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 2-x \rvert 
                             
l.509 ...t \frac32$, so $\frac1{\lvert 2-x \rvert}
                                                   \lt 2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.509 ...rac32$, so $\frac1{\lvert 2-x \rvert} \lt
                                                   2$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 509--510
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 509--510
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (11.38614pt too wide) in paragraph at lines 509--510
\OT1/cmr/m/n/10 1[]$
 []


Overfull \hbox (19.16394pt too wide) in paragraph at lines 509--510
[]\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (12.1014pt too wide) in paragraph at lines 509--510
\OML/cmm/m/it/10 x[]$
 []


Overfull \hbox (22.17088pt too wide) in paragraph at lines 509--510
[]\OT1/cmr/m/n/10 2$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.511             T
                   hat tells me $\left\lvert f(x)-2 \right\rvert = \frac{3\l...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.511             That tells me $\left\lvert
                                             f(x)-2 \right\rvert = \frac{3\l...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.511             That tells me $\left\lvert f
                                              (x)-2 \right\rvert = \frac{3\l...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.511 ...tells me $\left\lvert f(x)-2 \right\rvert
                                                   = \frac{3\lvert x-1 \rver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.511 ...lls me $\left\lvert f(x)-2 \right\rvert =
                                                   \frac{3\lvert x-1 \rvert}...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> 3\lvert 
                   x-1 \rvert 
l.511 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt 6\lvert x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> 3\lvert x-1 \rvert 
                              
l.511 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt 6\lvert x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 
                  2-x \rvert 
l.511 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt 6\lvert x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \lvert 2-x \rvert 
                             
l.511 ...ac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}
                                                   \lt 6\lvert x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.511 ...\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt
                                                   6\lvert x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.511 ...-1 \rvert}{\lvert 2-x \rvert} \lt 6\lvert
                                                   x-1 \rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.511 ...\lvert 2-x \rvert} \lt 6\lvert x-1 \rvert
                                                  $.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 511--512
[]
 []


Overfull \hbox (49.64299pt too wide) in paragraph at lines 511--512
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (39.65009pt too wide) in paragraph at lines 511--512
[]\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 511--512
\OT1/cmr/m/n/10 1$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.513             N
                   ow to achieve $\left\lvert f(x)-2 \right\rvert \lt \varep...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.513             Now to achieve $\left\lvert
                                              f(x)-2 \right\rvert \lt \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.513             Now to achieve $\left\lvert f
                                               (x)-2 \right\rvert \lt \varep...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.513 ... achieve $\left\lvert f(x)-2 \right\rvert
                                                   \lt \varepsilon$, all I n...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.513 ...ieve $\left\lvert f(x)-2 \right\rvert \lt
                                                   \varepsilon$, all I need ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.513 ...lvert f(x)-2 \right\rvert \lt \varepsilon
                                                  $, all I need is $6\lvert ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.513 ... \lt \varepsilon$, all I need is $6\lvert
                                                   x-1 \rvert \lt \varepsilo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.513 ...silon$, all I need is $6\lvert x-1 \rvert
                                                   \lt \varepsilon$; equival...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.513 ...n$, all I need is $6\lvert x-1 \rvert \lt
                                                   \varepsilon$; equivalentl...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no ; in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.513 ...rt \lt \varepsilon$; equivalently $\lvert
                                                   x-1 \rvert \lt \frac\vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.513 ...epsilon$; equivalently $\lvert x-1 \rvert
                                                   \lt \frac\varepsilon6$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.513 ...lon$; equivalently $\lvert x-1 \rvert \lt
                                                   \frac\varepsilon6$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 513--514
[]
 []


Overfull \hbox (45.41728pt too wide) in paragraph at lines 513--514
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 513--514
\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.6632pt too wide) in paragraph at lines 513--514
\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 "$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 513--514
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (11.38614pt too wide) in paragraph at lines 513--514
\OT1/cmr/m/n/10 1[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.515             T
                   o make all this work, I need both $\lvert x-1 \rvert \lt ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.515 ...o make all this work, I need both $\lvert
                                                   x-1 \rvert \lt \frac12$ a...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.515 ...this work, I need both $\lvert x-1 \rvert
                                                   \lt \frac12$ and $\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.515 ... work, I need both $\lvert x-1 \rvert \lt
                                                   \frac12$ and $\lvert x-1 ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.515 ...lvert x-1 \rvert \lt \frac12$ and $\lvert
                                                   x-1 \rvert \lt \frac\vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.515 ...rvert \lt \frac12$ and $\lvert x-1 \rvert
                                                   \lt \frac\varepsilon6$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.515 ...t \lt \frac12$ and $\lvert x-1 \rvert \lt
                                                   \frac\varepsilon6$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 515--516
[]
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 515--516
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (11.38614pt too wide) in paragraph at lines 515--516
\OT1/cmr/m/n/10 1[]$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 515--516
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (11.38614pt too wide) in paragraph at lines 515--516
\OT1/cmr/m/n/10 1[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.517             S
                   o choose $\delta = \min\\{\frac\varepsilon6,\frac12\\}$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.520             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.520             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.521             #
                   ## Why not input at the limit point?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.521             ##
                    # Why not input at the limit point?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.521             ###
                      Why not input at the limit point?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no W in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ? in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.522             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.522             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.522             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.522             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.523 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 517--524
[]
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (18.33334pt too wide) in paragraph at lines 517--524
[] [] 
 []


Overfull \hbox (17.21666pt too wide) in paragraph at lines 517--524
[]$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 517--524
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.525             f
                   or all $\varepsilon>0$, &nbsp; $\exists \delta>0$ such th...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.525             for all $\varepsilon>0$, &
                                            nbsp; $\exists \delta>0$ such th...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
! Misplaced alignment tab character &.
l.525 ...0$, &nbsp; $\exists \delta>0$ such that &
                                                  nbsp; $0 \lt \lvert x-a \r...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
! Undefined control sequence.
l.525 ...\exists \delta>0$ such that &nbsp; $0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.525 ... \delta>0$ such that &nbsp; $0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.525 ...such that &nbsp; $0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$$\lve...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.525 ... that &nbsp; $0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$$\lvert f...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.525 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $$\lvert f(x)-\ell \rvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \lvert 
                       
l.525 ...rt x-a \rvert \lt \delta \implies$$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.525 ... \delta \implies$$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 525--526
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 525--526
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 525--526
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 525--526
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 525--526
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 525--526
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (37.35193pt too wide) in paragraph at lines 525--526
\OML/cmm/m/it/10 a^^N$$f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \O
MS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 525--526
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 525--526
\OML/cmm/m/it/10 "$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.527             <
                   div style="display:flex;flex-direction:row;width:100%">
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 527--529
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.530             W
                   hy $0 \lt \lvert x-a \rvert \lt \delta$ not just $\lvert ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no W in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
l.530             Why $0 \lt
                             \lvert x-a \rvert \lt \delta$ not just $\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.530             Why $0 \lt \lvert
                                    x-a \rvert \lt \delta$ not just $\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.530             Why $0 \lt \lvert x-a \rvert
                                               \lt \delta$ not just $\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.530             Why $0 \lt \lvert x-a \rvert \lt
                                                   \delta$ not just $\lvert ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.530 ...t x-a \rvert \lt \delta$ not just $\lvert
                                                   x-a \rvert \lt \delta$?
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.530 ...t \lt \delta$ not just $\lvert x-a \rvert
                                                   \lt \delta$?
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.530 ...t \delta$ not just $\lvert x-a \rvert \lt
                                                   \delta$?
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no ? in font nullfont!

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[]
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Overfull \hbox (18.49309pt too wide) in paragraph at lines 530--531
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


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\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 530--531
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 530--531
\OML/cmm/m/it/10 a^^N$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.532             C
                   onsider $\chi\_{\\{3\\}}:\mathbb R\to\mathbb R$, the indi...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
! Undefined control sequence.
l.532 ...        Consider $\chi\_{\\{3\\}}:\mathbb
                                                   R\to\mathbb R$, the indic...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.532 ...ider $\chi\_{\\{3\\}}:\mathbb R\to\mathbb
                                                   R$, the indicator functio...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

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[]
 []


Overfull \hbox (16.81242pt too wide) in paragraph at lines 532--535
\OML/cmm/m/it/10 ^^_[][] \OT1/cmr/m/n/10 :
 []


Overfull \hbox (20.44788pt too wide) in paragraph at lines 532--535
\OML/cmm/m/it/10 R \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 532--535
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 532--535
[]$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 532--535
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 532--535
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 532--535
\OT1/cmr/m/n/10 3$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.536             !
                   [Desmos plot of y=indicator function for set {3} of x](gf...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.536 ...of x](gfx/chi3.png "Desmos plot of y=\chi
                                                  _{3}(x)")
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.539 
      
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


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[]
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Overfull \hbox (43.68054pt too wide) in paragraph at lines 536--539
\OML/cmm/m/it/10 ^^_[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )") \O
ML/cmm/m/it/10 <
 []


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\OML/cmm/m/it/10 =div ><
 []


Overfull \hbox (29.41153pt too wide) in paragraph at lines 536--539
\OML/cmm/m/it/10 =div >$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.540             W
                   e want limits to be defined so that $\displaystyle \lim\_...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no W in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.543             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.543             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.544             #
                   ### Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.544             ##
                    ## Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.544             ###
                     # Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.544             ####
                       Definition
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.545 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

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[]
 []


Overfull \hbox (77.13179pt too wide) in paragraph at lines 540--546
[] [][]\OML/cmm/m/it/10 ^^_[][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/
10 ) =
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 540--546
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 540--546
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 540--546
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 540--546
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 540--546
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 540--546
\OML/cmm/m/it/10 a$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.547             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.547 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.547 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.547 ..."color:var(--uswds-green-cool-20)">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.547 ...var(--uswds-green-cool-20)">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.547 ...-green-cool-20)">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.547 ...en-cool-20)">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.547 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.547 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.547 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 547--548
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 547--548
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 547--548
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 547--548
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 547--548
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 547--548
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 547--548
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 547--548
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 547--548
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 547--548
\OML/cmm/m/it/10 "$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.549             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.549             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.549             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.549             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.549             #### T
                        heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.551             #
                   ### Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.551             ##
                    ## Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.551             ###
                     # Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.551             ####
                       Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 549--553
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 549--553
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (24.75pt too wide) in paragraph at lines 549--553
\OML/cmm/m/it/10 ^^_[][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (29.30559pt too wide) in paragraph at lines 549--553
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =
 []


Overfull \hbox (10.00003pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 12$
 []


Overfull \hbox (62.5695pt too wide) in paragraph at lines 549--553
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (4.72223pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 ?$
 []


Overfull \hbox (29.30559pt too wide) in paragraph at lines 549--553
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =
 []


Overfull \hbox (25.55563pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 5(2) +
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (34.59026pt too wide) in paragraph at lines 549--553
\OML/cmm/m/it/10 ^^_[][]\OT1/cmr/m/n/10 (1) =
 []


Overfull \hbox (17.77783pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 10 +
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (15.55553pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 1 =
 []


Overfull \hbox (10.00003pt too wide) in paragraph at lines 549--553
\OT1/cmr/m/n/10 12$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.554             Y
                   our turn!
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no Y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ! in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.556             #
                   ### How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.556             ##
                    ## How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.556             ###
                     # How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.556             ####
                       How to prove convergence if you know the limit
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no H in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.560             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.560             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.561             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.561             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.561             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.561             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.563             #
                   ### Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.563             ##
                    ## Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.563             ###
                     # Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.563             ####
                       Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no R in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 554--565
[]
 []


Overfull \hbox (11.66661pt too wide) in paragraph at lines 554--565
[]$
 []


Overfull \hbox (4.8229pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 "$
 []


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\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


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\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
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\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
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\OML/cmm/m/it/10 ^^_[][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (29.30559pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =
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\OT1/cmr/m/n/10 12$
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Overfull \hbox (62.5695pt too wide) in paragraph at lines 554--565
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (4.72223pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 ?$
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Overfull \hbox (16.27078pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^Y
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Overfull \hbox (5.00002pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 2$
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Overfull \hbox (16.27078pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =
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Overfull \hbox (5.00002pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 2$
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Overfull \hbox (30.02084pt too wide) in paragraph at lines 554--565
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^Y
 []


Overfull \hbox (25.55563pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 5(2) +
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\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
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Overfull \hbox (15.55553pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 0 =
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\OT1/cmr/m/n/10 13$
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Overfull \hbox (10.00003pt too wide) in paragraph at lines 554--565
\OT1/cmr/m/n/10 13$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.566             I
                    want to make
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.567             $\left\lvert
                               f(x)-13 \right\rvert \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.567             $\left\lvert f
                                (x)-13 \right\rvert \lt \varepsilon$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.567 ...        $\left\lvert f(x)-13 \right\rvert
                                                   \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.567 ...    $\left\lvert f(x)-13 \right\rvert \lt
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.567 ...vert f(x)-13 \right\rvert \lt \varepsilon
                                                  $.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 566--570
[]
 []


Overfull \hbox (50.4173pt too wide) in paragraph at lines 566--570
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (16.27078pt too wide) in paragraph at lines 566--570
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =
 []


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\OT1/cmr/m/n/10 2$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 566--570
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


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\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


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\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 566--570
\OT1/cmr/m/n/10 0$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.571             I
                    calculate $\left\lvert f(x)-13 \right\rvert = \left\lver...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.571             I calculate $\left\lvert
                                           f(x)-13 \right\rvert = \left\lver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.571             I calculate $\left\lvert f
                                            (x)-13 \right\rvert = \left\lver...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.571 ...lculate $\left\lvert f(x)-13 \right\rvert
                                                   = \left\lvert 5x+3-13 \ri...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.571 ...ulate $\left\lvert f(x)-13 \right\rvert =
                                                   \left\lvert 5x+3-13 \righ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.571 ...\lvert f(x)-13 \right\rvert = \left\lvert
                                                   5x+3-13 \right\rvert = 5\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   5
l.571 ...vert f(x)-13 \right\rvert = \left\lvert 5
                                                  x+3-13 \right\rvert = 5\le...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.571 ...\rvert = \left\lvert 5x+3-13 \right\rvert
                                                   = 5\left\lvert x-2 \right...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.571 ...vert = \left\lvert 5x+3-13 \right\rvert =
                                                   5\left\lvert x-2 \right\r...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.571 ...lvert 5x+3-13 \right\rvert = 5\left\lvert
                                                   x-2 \right\rvert$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   x
l.571 ...ert 5x+3-13 \right\rvert = 5\left\lvert x
                                                  -2 \right\rvert$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.571 ...ght\rvert = 5\left\lvert x-2 \right\rvert
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   $
l.571 ...ht\rvert = 5\left\lvert x-2 \right\rvert$
                                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


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\OT1/cmr/m/n/10 5 []$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.573             a
                   nd I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \de...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
l.573             and I'm allowed to insist $0 \lt
                                                   \lvert x-2 \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.573 ...  and I'm allowed to insist $0 \lt \lvert
                                                   x-2 \rvert \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.573 ...llowed to insist $0 \lt \lvert x-2 \rvert
                                                   \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.573 ...ed to insist $0 \lt \lvert x-2 \rvert \lt
                                                   \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.574             Now to achieve $\left\lvert
                                              f(x)-13 \right\rvert \lt \vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.574             Now to achieve $\left\lvert f
                                               (x)-13 \right\rvert \lt \vare...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.574 ...achieve $\left\lvert f(x)-13 \right\rvert
                                                   \lt \varepsilon$, all I n...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.574 ...eve $\left\lvert f(x)-13 \right\rvert \lt
                                                   \varepsilon$, all I need ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.574 ...vert f(x)-13 \right\rvert \lt \varepsilon
                                                  $, all I need is $5\lvert ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.574 ... \lt \varepsilon$, all I need is $5\lvert
                                                   x-2 \rvert \lt \varepsilo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.574 ...silon$, all I need is $5\lvert x-2 \rvert
                                                   \lt \varepsilon$; equival...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.574 ...n$, all I need is $5\lvert x-2 \rvert \lt
                                                   \varepsilon$; equivalentl...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no ; in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.574 ...rt \lt \varepsilon$; equivalently $\lvert
                                                   x-2 \rvert \lt \frac\vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.574 ...epsilon$; equivalently $\lvert x-2 \rvert
                                                   \lt \frac\varepsilon5$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.574 ...lon$; equivalently $\lvert x-2 \rvert \lt
                                                   \frac\varepsilon5$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.578             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.578             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.579             #
                   ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.579             ##
                    ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.579             ###
                     # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.579             ####
                       Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.581             #
                   ### Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.581             ##
                    ## Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.581             ###
                     # Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.581             ####
                       Rough work
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no R in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

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Overfull \hbox (18.49309pt too wide) in paragraph at lines 573--583
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
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\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ^^N$
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[] \OML/cmm/m/it/10 "$
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\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 "$
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\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
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\OT1/cmr/m/n/10 2[]$
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\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
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[]$
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Overfull \hbox (30.02084pt too wide) in paragraph at lines 573--583
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
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\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
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\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
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\OML/cmm/m/it/10 ^^_[][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
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\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =
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\OT1/cmr/m/n/10 12$
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y/m/n/10 ^^Y
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.584             I
                    want to make
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.585             $\left\lvert
                               f(x)-13 \right\rvert \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.585             $\left\lvert f
                                (x)-13 \right\rvert \lt \varepsilon$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.585 ...        $\left\lvert f(x)-13 \right\rvert
                                                   \lt \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.585 ...    $\left\lvert f(x)-13 \right\rvert \lt
                                                   \varepsilon$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.585 ...vert f(x)-13 \right\rvert \lt \varepsilon
                                                  $.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 584--588
[]
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Overfull \hbox (50.4173pt too wide) in paragraph at lines 584--588
[] \OML/cmm/m/it/10 "$
 []


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 []


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\OT1/cmr/m/n/10 2$
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 []


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\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


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\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^@
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\OT1/cmr/m/n/10 0$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.589             I
                    calculate $\left\lvert f(x)-13 \right\rvert = \left\lver...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.589             I calculate $\left\lvert
                                           f(x)-13 \right\rvert = \left\lver...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.589             I calculate $\left\lvert f
                                            (x)-13 \right\rvert = \left\lver...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.589 ...lculate $\left\lvert f(x)-13 \right\rvert
                                                   = \left\lvert 5x+3-13 \ri...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.589 ...ulate $\left\lvert f(x)-13 \right\rvert =
                                                   \left\lvert 5x+3-13 \righ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.589 ...\lvert f(x)-13 \right\rvert = \left\lvert
                                                   5x+3-13 \right\rvert = 5\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   5
l.589 ...vert f(x)-13 \right\rvert = \left\lvert 5
                                                  x+3-13 \right\rvert = 5\le...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.589 ...\rvert = \left\lvert 5x+3-13 \right\rvert
                                                   = 5\left\lvert x-2 \right...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.589 ...vert = \left\lvert 5x+3-13 \right\rvert =
                                                   5\left\lvert x-2 \right\r...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.589 ...lvert 5x+3-13 \right\rvert = 5\left\lvert
                                                   x-2 \right\rvert$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   x
l.589 ...ert 5x+3-13 \right\rvert = 5\left\lvert x
                                                  -2 \right\rvert$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.589 ...ght\rvert = 5\left\lvert x-2 \right\rvert
                                                  $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   $
l.589 ...ht\rvert = 5\left\lvert x-2 \right\rvert$
                                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


Overfull \hbox (20.0pt too wide) in paragraph at lines 589--590
[]
 []


Overfull \hbox (54.643pt too wide) in paragraph at lines 589--590
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (63.11511pt too wide) in paragraph at lines 589--590
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (32.00406pt too wide) in paragraph at lines 589--590
\OT1/cmr/m/n/10 5 []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.591             a
                   nd I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \de...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
l.591             and I'm allowed to insist $0 \lt
                                                   \lvert x-2 \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.591 ...  and I'm allowed to insist $0 \lt \lvert
                                                   x-2 \rvert \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.591 ...llowed to insist $0 \lt \lvert x-2 \rvert
                                                   \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.591 ...ed to insist $0 \lt \lvert x-2 \rvert \lt
                                                   \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.592             Now to achieve $\left\lvert
                                              f(x)-13 \right\rvert \lt \vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.592             Now to achieve $\left\lvert f
                                               (x)-13 \right\rvert \lt \vare...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.592 ...achieve $\left\lvert f(x)-13 \right\rvert
                                                   \lt \varepsilon$, all I n...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.592 ...eve $\left\lvert f(x)-13 \right\rvert \lt
                                                   \varepsilon$, all I need ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \varepsilon 
l.592 ...vert f(x)-13 \right\rvert \lt \varepsilon
                                                  $, all I need is $5\lvert ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.592 ... \lt \varepsilon$, all I need is $5\lvert
                                                   x-2 \rvert \lt \varepsilo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.592 ...silon$, all I need is $5\lvert x-2 \rvert
                                                   \lt \varepsilon$; equival...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.592 ...n$, all I need is $5\lvert x-2 \rvert \lt
                                                   \varepsilon$; equivalentl...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no ; in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.592 ...rt \lt \varepsilon$; equivalently $\lvert
                                                   x-2 \rvert \lt \frac\vare...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.592 ...epsilon$; equivalently $\lvert x-2 \rvert
                                                   \lt \frac\varepsilon5$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.592 ...lon$; equivalently $\lvert x-2 \rvert \lt
                                                   \frac\varepsilon5$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.594             #
                   ### Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.594             ##
                    ## Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.594             ###
                     # Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.594             ####
                       Proof
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
! Undefined control sequence.
l.596             If $0 \lt
                            \lvert x-2 \rvert \lt \delta$, then $\left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.596             If $0 \lt \lvert
                                   x-2 \rvert \lt \delta$, then $\left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.596             If $0 \lt \lvert x-2 \rvert
                                              \lt \delta$, then $\left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.596             If $0 \lt \lvert x-2 \rvert \lt
                                                  \delta$, then $\left\lvert...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.596 ...x-2 \rvert \lt \delta$, then $\left\lvert
                                                   f(x)-13 \right\rvert = \l...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.596 ...2 \rvert \lt \delta$, then $\left\lvert f
                                                  (x)-13 \right\rvert = \lef...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.596 ...$, then $\left\lvert f(x)-13 \right\rvert
                                                   = \left\lvert 5x+3-\chi\_...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.596 ... then $\left\lvert f(x)-13 \right\rvert =
                                                   \left\lvert 5x+3-\chi\_{\...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.596 ...\lvert f(x)-13 \right\rvert = \left\lvert
                                                   5x+3-\chi\_{\\{2\\}}(x)-1...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   5
l.596 ...vert f(x)-13 \right\rvert = \left\lvert 5
                                                  x+3-\chi\_{\\{2\\}}(x)-13 ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.596 ...t 5x+3-\chi\_{\\{2\\}}(x)-13 \right\rvert
                                                   = 5 \lvert x-2 \rvert\lt5...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   =
l.596 ...5x+3-\chi\_{\\{2\\}}(x)-13 \right\rvert =
                                                   5 \lvert x-2 \rvert\lt5\f...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.596 ...\_{\\{2\\}}(x)-13 \right\rvert = 5 \lvert
                                                   x-2 \rvert\lt5\frac\varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.596 ...(x)-13 \right\rvert = 5 \lvert x-2 \rvert
                                                  \lt5\frac\varepsilon5=\var...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.596 ...-13 \right\rvert = 5 \lvert x-2 \rvert\lt
                                                  5\frac\varepsilon5=\vareps...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!
! Misplaced alignment tab character &.
l.596 ...vert\lt5\frac\varepsilon5=\varepsilon$. &
                                                  nbsp; □
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!

! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.596 ...frac\varepsilon5=\varepsilon$. &nbsp; □
                                                  
You may provide a definition with
\DeclareUnicodeCharacter 

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.599             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.599             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 591--602
[]
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (50.4173pt too wide) in paragraph at lines 591--602
[] \OML/cmm/m/it/10 "$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.6632pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 "$
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 591--602
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (11.38614pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 2[]$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 591--602
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (6.38612pt too wide) in paragraph at lines 591--602
[]$
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 591--602
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 591--602
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (6.38612pt too wide) in paragraph at lines 591--602
[]$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ^^N$
 []


Overfull \hbox (54.643pt too wide) in paragraph at lines 591--602
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (100.08725pt too wide) in paragraph at lines 591--602
[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (26.94167pt too wide) in paragraph at lines 591--602
\OT1/cmr/m/n/10 25[] =
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 591--602
\OML/cmm/m/it/10 "$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.603               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 603--604
[]
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 603--604
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 603--604
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.605               !
                     [Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgn...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 605--606
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.607               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 607--610
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.611               $
                     \displaystyle f(2)=$ <!-- .element: style="text-align:r...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (49.30559pt too wide) in paragraph at lines 611--612
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.613               $
                     \displaystyle\lim\_{x\to3}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 613--614
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.615               $
                     \displaystyle\lim\_{x\to2}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 615--618
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.620             <
                   /div>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.624             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.624             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 620--627
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.628               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 628--629
[]
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 628--629
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 628--629
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.630               !
                     [Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgn...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
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Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
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Missing character: There is no - in font nullfont!
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Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 630--631
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.632               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
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Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
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Missing character: There is no : in font nullfont!
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Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 632--635
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.636               $
                     \displaystyle f(2)=$ <!-- .element: style="text-align:r...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no - in font nullfont!
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Overfull \hbox (49.30559pt too wide) in paragraph at lines 636--637
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.638               $
                     \displaystyle\lim\_{x\to3}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 638--639
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.640               $
                     \displaystyle\lim\_{x\to2}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
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Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
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Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
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Missing character: There is no . in font nullfont!
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Missing character: There is no x in font nullfont!
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Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 640--643
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.644               $
                     \displaystyle 2 \phantom{f(2)}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (43.75009pt too wide) in paragraph at lines 644--645
[]$\OT1/cmr/m/n/10 2[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.646               $
                     \displaystyle 4 \phantom{\lim\_{x\to3}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (66.8263pt too wide) in paragraph at lines 646--647
[]$\OT1/cmr/m/n/10 4[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.648               u
                     ndefined $\displaystyle \phantom{\lim\_{x\to2}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.653             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.653             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 648--656
[]
 []


Overfull \hbox (41.82628pt too wide) in paragraph at lines 648--656
[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.657               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 657--658
[]
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 657--658
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 657--658
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.659               !
                     [Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgn...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 659--660
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.661               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 661--664
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.665               $
                     \displaystyle f(2)=$ <!-- .element: style="text-align:r...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (49.30559pt too wide) in paragraph at lines 665--666
[]$\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.667               $
                     \displaystyle\lim\_{x\to3}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 667--668
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.669               $
                     \displaystyle\lim\_{x\to2}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 669--672
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.673               $
                     \displaystyle 2 \phantom{f(2)}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (43.75009pt too wide) in paragraph at lines 673--674
[]$\OT1/cmr/m/n/10 2[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.675               $
                     \displaystyle 4 \phantom{\lim\_{x\to3}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (66.8263pt too wide) in paragraph at lines 675--676
[]$\OT1/cmr/m/n/10 4[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.677               u
                     ndefined $\displaystyle \phantom{\lim\_{x\to2}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 677--678
[]
 []


Overfull \hbox (41.82628pt too wide) in paragraph at lines 677--678
[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.679               b
                     ut it "should" be $1$ or $3$, depending on which side w...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.684             #
                   # Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.684             ##
                     Limits of real functions
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 679--687
[]
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 679--687
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 679--687
\OT1/cmr/m/n/10 3$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.688               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 688--689
[]
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 688--689
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 688--689
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.690               !
                     [Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgn...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 690--691
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.692               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 692--695
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.696               #
                     ### Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.696               ##
                      ## Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.696               ###
                       # Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.696               ####
                         Notation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.696               #### N
                          otation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no O in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 696--698
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.699               *
                     the limit from the left* $\displaystyle \lim\_{x\to a^-...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 699--700
[]
 []


Overfull \hbox (68.3275pt too wide) in paragraph at lines 699--700
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.701               *
                     the limit from the right* $\displaystyle \lim\_{x\to a^...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 701--702
[]
 []


Overfull \hbox (68.2164pt too wide) in paragraph at lines 701--702
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 

 []

! You can't use `macro parameter character #' in vertical mode.
l.703               #
                     ### Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.703               ##
                      ## Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.703               ###
                       # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.703               ####
                         Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.703               #### E
                          xample
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 703--708
[]
 []


Overfull \hbox (5.97226pt too wide) in paragraph at lines 703--708
\OML/cmm/m/it/10 f$
 []


Overfull \hbox (78.59714pt too wide) in paragraph at lines 703--708
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 703--708
\OT1/cmr/m/n/10 1$
 []


Overfull \hbox (78.48604pt too wide) in paragraph at lines 703--708
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 703--708
\OT1/cmr/m/n/10 3$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.709               #
                     ### Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.709               ##
                      ## Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.709               ###
                       # Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.709               ####
                         Theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.709               #### T
                          heorem
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 709--713
[]
 []


Overfull \hbox (61.57748pt too wide) in paragraph at lines 709--713
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (78.88301pt too wide) in paragraph at lines 709--713
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (68.2164pt too wide) in paragraph at lines 709--713
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.714             <
                   /div>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.718             #
                   # Definitions of one sided limits
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.718             ##
                     Definitions of one sided limits
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.719             #
                   ### Definition (recall; two sided limit)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.719             ##
                    ## Definition (recall; two sided limit)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.719             ###
                     # Definition (recall; two sided limit)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.719             ####
                       Definition (recall; two sided limit)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.720 ...)$ *converges* to *limit* $\ell\in\mathbb
                                                   R$ as $x\to a$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

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[]
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\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
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\OML/cmm/m/it/10 R$
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\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
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\OML/cmm/m/it/10 a$
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.722             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.722 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.722 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.722 ...--uswds-green-cool-20);background-color:#
                                                  505050">$0 \lt \lvert x-a ...
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.722 ...cool-20);background-color:#505050">$0 \lt
                                                   \lvert x-a \rvert \lt \de...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.722 ...);background-color:#505050">$0 \lt \lvert
                                                   x-a \rvert \lt \delta \im...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.722 ...d-color:#505050">$0 \lt \lvert x-a \rvert
                                                   \lt \delta \implies$</spa...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.722 ...lor:#505050">$0 \lt \lvert x-a \rvert \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.722 ...\lt \lvert x-a \rvert \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.722 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.722 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.723             #
                   ### Definition (limit from the left)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.723             ##
                    ## Definition (limit from the left)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.723             ###
                     # Definition (limit from the left)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.723             ####
                       Definition (limit from the left)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.724 ... to *limit from the left* $\ell\in\mathbb
                                                   R$ as $x\to a^-$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 722--725
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 722--725
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 722--725
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 722--725
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 722--725
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.1088pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 a^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 "$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (12.0359pt too wide) in paragraph at lines 722--725
\OML/cmm/m/it/10 a[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.726             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.726 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.726 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.726 ...--uswds-green-cool-20);background-color:#
                                                  505050">$0 \lt a-x \lt \de...
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.726 ...cool-20);background-color:#505050">$0 \lt
                                                   a-x \lt \delta \implies$<...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.726 ...;background-color:#505050">$0 \lt a-x \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.726 ...r:#505050">$0 \lt a-x \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.726 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.726 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.727             #
                   ### Definition (limit from the right)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.727             ##
                    ## Definition (limit from the right)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.727             ###
                     # Definition (limit from the right)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.727             ####
                       Definition (limit from the right)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no * in font nullfont!
! Undefined control sequence.
l.728 ...to *limit from the right* $\ell\in\mathbb
                                                   R$ as $x\to a^+$ if
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 726--729
[]
 []


Overfull \hbox (15.21869pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 " >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 726--729
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (20.93399pt too wide) in paragraph at lines 726--729
\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 726--729
\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (18.0637pt too wide) in paragraph at lines 726--729
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (10.53818pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 x^^N$
 []


Overfull \hbox (27.24313pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 ` <
 []


Overfull \hbox (4.66318pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 "$
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.61108pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 ` \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (7.67015pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 R$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (11.9248pt too wide) in paragraph at lines 726--729
\OML/cmm/m/it/10 a[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.730             <
                   span style="color:var(--uswds-red-cool-20)">for all $\var...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no , in font nullfont!
! Misplaced alignment tab character &.
l.730 ...ol-20)">for all $\varepsilon>0$</span>, &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
! Misplaced alignment tab character &.
l.730 ...0)">$\exists \delta>0$ such that</span> &
                                                  nbsp; <span style="color:v...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.730 ...--uswds-green-cool-20);background-color:#
                                                  505050">$0 \lt x-a \lt \de...
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
l.730 ...cool-20);background-color:#505050">$0 \lt
                                                   x-a \lt \delta \implies$<...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.730 ...;background-color:#505050">$0 \lt x-a \lt
                                                   \delta \implies$</span> <...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.730 ...r:#505050">$0 \lt x-a \lt \delta \implies
                                                  $</span> <span style="colo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
! Undefined control sequence.
<recently read> \lvert 
                       
l.730 ...tyle="color:var(--uswds-gold-20)">$\lvert
                                                   f(x)-\ell \rvert < \varep...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.730 ...-uswds-gold-20)">$\lvert f(x)-\ell \rvert
                                                   < \varepsilon$</span>.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.733             #
                   # &#128247; Algebra of limits
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.733             ##
                     &#128247; Algebra of limits
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Misplaced alignment tab character &.
l.733             ## &
                      #128247; Algebra of limits
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! You can't use `macro parameter character #' in horizontal mode.
l.733             ## &#
                       128247; Algebra of limits
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no 1 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 8 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
! Undefined control sequence.
l.734 ...{x\to a}g(x)$ both exist and $c\in\mathbb
                                                   R$, then
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 730--739
[]
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\OML/cmm/m/it/10 " >
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\OT1/cmr/m/n/10 0$
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\OMS/cmsy/m/n/10 9\OML/cmm/m/it/10 ^^N >
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\OT1/cmr/m/n/10 0$
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\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
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\OML/cmm/m/it/10 a^^N$
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Overfull \hbox (27.24313pt too wide) in paragraph at lines 730--739
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cms
y/m/n/10 ^^@
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\OML/cmm/m/it/10 ` <
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\OML/cmm/m/it/10 "$
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[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
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\OML/cmm/m/it/10 c \OMS/cmsy/m/n/10 2
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Overfull \hbox (7.67015pt too wide) in paragraph at lines 730--739
\OML/cmm/m/it/10 R$
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Overfull \hbox (73.24419pt too wide) in paragraph at lines 730--739
[] [][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\O
T1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^F
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\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )) =
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[] \OMS/cmsy/m/n/10 ^^F
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[]$
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Overfull \hbox (98.53233pt too wide) in paragraph at lines 730--739
[] [][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\O
T1/cmr/m/n/10 )\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/
n/10 )) =
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[] []$
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[] [][] [] \OT1/cmr/m/n/10 =
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[]$
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Overfull \hbox (71.2892pt too wide) in paragraph at lines 730--739
[] [][]\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \
OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =
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\OT1/cmr/m/n/10 0$
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[] [][]\OML/cmm/m/it/10 cf\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ))
 =
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\OML/cmm/m/it/10 c []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.740             I
                   n each of these, $x\to a$ can be replaced by $x\to a^\pm$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.743             #
                   # &#128247; Squeeze theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.743             ##
                     &#128247; Squeeze theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Misplaced alignment tab character &.
l.743             ## &
                      #128247; Squeeze theorem
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! You can't use `macro parameter character #' in horizontal mode.
l.743             ## &#
                       128247; Squeeze theorem
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no 1 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 8 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
! Undefined control sequence.
l.744 ...$, and, for all $x\in I$, $f(x) \leqslant
                                                   g(x) \leqslant h(x)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.744 ... $x\in I$, $f(x) \leqslant g(x) \leqslant
                                                   h(x)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 740--745
[]
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (18.493pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
 []


Overfull \hbox (12.0359pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 a[]$
 []


Overfull \hbox (25.19557pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 f; g; h$
 []


Overfull \hbox (5.18054pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 I$
 []


Overfull \hbox (14.73029pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (5.18054pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 I$
 []


Overfull \hbox (15.15967pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (5.18054pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 I$
 []


Overfull \hbox (57.34152pt too wide) in paragraph at lines 740--745
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML/cmm/
m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 h
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.746             I
                   f $\displaystyle\lim\_{x\to a}f(x) = \ell = \displaystyle...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 746--747
[]
 []


Overfull \hbox (72.133pt too wide) in paragraph at lines 746--747
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (14.7222pt too wide) in paragraph at lines 746--747
\OML/cmm/m/it/10 ` \OT1/cmr/m/n/10 =
 []


Overfull \hbox (61.3668pt too wide) in paragraph at lines 746--747
[] [][]\OML/cmm/m/it/10 h\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 

 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.748             t
                   hen also $\displaystyle\lim\_{x\to a}g(x)=\ell$.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 748--749
[]
 []


Overfull \hbox (71.2892pt too wide) in paragraph at lines 748--749
[] [][]\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (4.16669pt too wide) in paragraph at lines 748--749
\OML/cmm/m/it/10 `$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.750             T
                   here are also versions for one sided limits.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.753             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.753             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 750--756
[]
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.757               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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Overfull \hbox (20.0pt too wide) in paragraph at lines 757--758
[]
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Overfull \hbox (15.81711pt too wide) in paragraph at lines 757--758
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 757--758
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.759               !
                     [Desmos plot of y=x plus sgn of x minus 2 plus 2x minus...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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Overfull \hbox (20.0pt too wide) in paragraph at lines 759--760
[]
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.761               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 761--764
[]
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.765               $
                     \displaystyle \phantom{\left(\frac52\right)}f(x) =$ <!-...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Overfull \hbox (72.14314pt too wide) in paragraph at lines 765--766
[]$[]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =$
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.767               $
                     \displaystyle\lim\_{x\to3}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Overfull \hbox (91.84712pt too wide) in paragraph at lines 767--768
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.769               $
                     \displaystyle\lim\_{x\to2^-}f(x)=$ <!-- .element: style...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
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Overfull \hbox (98.59714pt too wide) in paragraph at lines 769--770
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.771               $
                     \displaystyle\lim\_{x\to2^+}f(x)=$ <!-- .element: style...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (98.48604pt too wide) in paragraph at lines 771--772
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.773               $
                     \displaystyle\lim\_{x\to2}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 8 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 773--776
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.777               $
                     \displaystyle H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (46.50694pt too wide) in paragraph at lines 777--778
[]$\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
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Overfull \hbox (16.66672pt too wide) in paragraph at lines 777--778
\OT1/cmr/m/n/10 2) +
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Overfull \hbox (13.49307pt too wide) in paragraph at lines 777--778
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (22.38199pt too wide) in paragraph at lines 777--778
\OT1/cmr/m/n/10 2(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
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Overfull \hbox (64.55968pt too wide) in paragraph at lines 777--778
\OT1/cmr/m/n/10 2) [] []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.779               $
                     \displaystyle ? \phantom{\lim\_{3}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (45.2778pt too wide) in paragraph at lines 779--780
[]$\OT1/cmr/m/n/10 ?[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.781               $
                     \displaystyle ? \phantom{\lim\_{2^-}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (52.02782pt too wide) in paragraph at lines 781--782
[]$\OT1/cmr/m/n/10 ?[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.783               $
                     \displaystyle ? \phantom{\lim\_{2^+}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (51.91672pt too wide) in paragraph at lines 783--784
[]$\OT1/cmr/m/n/10 ?[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.785               $
                     \displaystyle ? \phantom{\lim\_{2}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.790             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.790             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (45.2778pt too wide) in paragraph at lines 785--793
[]$\OT1/cmr/m/n/10 ?[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.794               P
                     lot of $y=f(x)$. <!-- .element: style="text-align:cente...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 794--795
[]
 []


Overfull \hbox (15.81711pt too wide) in paragraph at lines 794--795
\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 =
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 794--795
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.796               !
                     [Desmos plot of y=x plus sgn of x minus 2 plus 2x minus...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 796--797
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.798               <
                     !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Missing character: There is no - in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 798--801
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.802               $
                     \displaystyle \phantom{\left(\frac52\right)}f(x) =$ <!-...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Overfull \hbox (72.14314pt too wide) in paragraph at lines 802--803
[]$[]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.804               $
                     \displaystyle\lim\_{x\to3}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Overfull \hbox (91.84712pt too wide) in paragraph at lines 804--805
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.806               $
                     \displaystyle\lim\_{x\to2^-}f(x)=$ <!-- .element: style...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
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Overfull \hbox (98.59714pt too wide) in paragraph at lines 806--807
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.808               $
                     \displaystyle\lim\_{x\to2^+}f(x)=$ <!-- .element: style...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
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Overfull \hbox (98.48604pt too wide) in paragraph at lines 808--809
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.810               $
                     \displaystyle\lim\_{x\to2}f(x)=$ <!-- .element: style="...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
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Missing character: There is no - in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no > in font nullfont!

Overfull \hbox (91.84712pt too wide) in paragraph at lines 810--813
[]$[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 
) =$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.814               $
                     \displaystyle H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (46.50694pt too wide) in paragraph at lines 814--815
[]$\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (16.66672pt too wide) in paragraph at lines 814--815
\OT1/cmr/m/n/10 2) +
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 814--815
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (22.38199pt too wide) in paragraph at lines 814--815
\OT1/cmr/m/n/10 2(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (64.55968pt too wide) in paragraph at lines 814--815
\OT1/cmr/m/n/10 2) [] []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.816               $
                     \displaystyle 2 \phantom{\lim\_{3}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (45.55559pt too wide) in paragraph at lines 816--817
[]$\OT1/cmr/m/n/10 2[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.818               $
                     \displaystyle 1 \phantom{\lim\_{2^-}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (52.3056pt too wide) in paragraph at lines 818--819
[]$\OT1/cmr/m/n/10 1[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.820               $
                     \displaystyle 3 \phantom{\lim\_{2^+}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (52.1945pt too wide) in paragraph at lines 820--821
[]$\OT1/cmr/m/n/10 3[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.822               u
                     ndefined $\displaystyle \phantom{\lim\_{2}}$
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.827             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.827             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.830             #
                   ### Proof (direct from definition)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.830             ##
                    ## Proof (direct from definition)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.830             ###
                     # Proof (direct from definition)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.830             ####
                       Proof (direct from definition)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no F in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no y in font nullfont!
! Undefined control sequence.
l.831             For any $\varepsilon\gt
                                         0$, let $\delta=\frac\varepsilon3$,...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.831 ...ta=\frac\varepsilon3$, and suppose $0 \lt
                                                   x-2 \lt \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.831 ...\varepsilon3$, and suppose $0 \lt x-2 \lt
                                                   \delta$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 822--832
[]
 []


Overfull \hbox (20.55557pt too wide) in paragraph at lines 822--832
[]$
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 822--832
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (26.50694pt too wide) in paragraph at lines 822--832
\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (16.66672pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 2) +
 []


Overfull \hbox (13.49307pt too wide) in paragraph at lines 822--832
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (22.38199pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 2(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (64.55968pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 2) [] []$
 []


Overfull \hbox (78.48604pt too wide) in paragraph at lines 822--832
[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 3$
 []


Overfull \hbox (9.6632pt too wide) in paragraph at lines 822--832
\OML/cmm/m/it/10 "\OT1/cmr/m/n/10 0$
 []


Overfull \hbox (15.37842pt too wide) in paragraph at lines 822--832
\OML/cmm/m/it/10 ^^N \OT1/cmr/m/n/10 =
 []


Overfull \hbox (6.38612pt too wide) in paragraph at lines 822--832
[]$
 []


Overfull \hbox (18.49309pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (9.82292pt too wide) in paragraph at lines 822--832
\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ^^N$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.833             T
                   hen $x \gt 2$, so $H(x-2)=1$, so
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
! Undefined control sequence.
l.833             Then $x \gt
                              2$, so $H(x-2)=1$, so
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   \protect 
l.842             \end{aligned}\\]
                                  
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   \protect 
l.842             \end{aligned}\\]
                                  
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)


! LaTeX Error: Environment aligned undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.842             \end{aligned}\\]
                                  
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.842             \end{aligned}\\]
                                  
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.842             \end{aligned}\\]
                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

! Undefined control sequence.
<argument> \begin {aligned} \left \lvert 
                                         f(x) - 3 \right \rvert &= \left \lv...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> ...\left \lvert f(x) - 3 \right \rvert 
                                                  &= \left \lvert 1+x+2(x-2)...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   &
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Misplaced alignment tab character &.
<recently read> &
                 
l.842             \end{aligned}\\]
                                  
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
<argument> ...) - 3 \right \rvert &= \left \lvert 
                                                  1+x+2(x-2)\sin \left (\fra...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   1
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> ...ac 5{x-2}\right ) - 3 \right \rvert 
                                                  \\\\ &= \left \lvert x+2(x...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \let 
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Misplaced alignment tab character &.
<recently read> &
                 
l.842             \end{aligned}\\]
                                  
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
<argument> ... \right \rvert \\\\ &= \left \lvert 
                                                  x+2(x-2)\sin \left (\frac ...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   x
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> ...ac 5{x-2}\right ) - 2 \right \rvert 
                                                  \\\\ &= (x-2) \left \lvert...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \let 
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Misplaced alignment tab character &.
<recently read> &
                 
l.842             \end{aligned}\\]
                                  
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
<argument> ...t \rvert \\\\ &= (x-2) \left \lvert 
                                                  1 + 2\sin \left (\frac 5{x...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   1
l.842             \end{aligned}\\]
                                  
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
<argument> ...(\frac 5{x-2}\right ) \right \rvert 
                                                  \leqslant (x-2) 3 \\\\ &\l...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> ...-2}\right ) \right \rvert \leqslant 
                                                  (x-2) 3 \\\\ &\lt 3 \delta...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Misplaced alignment tab character &.
<recently read> &
                 
l.842             \end{aligned}\\]
                                  
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
<argument> ... \rvert \leqslant (x-2) 3 \\\\ &\lt 
                                                  3 \delta = 3\frac \varepsi...
l.842             \end{aligned}\\]
                                  
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.842             \end{aligned}\\]
                                  
You may provide a definition with
\DeclareUnicodeCharacter 


! LaTeX Error: \begin{document} ended by \end{aligned}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.842             \end{aligned}\\]
                                  
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.842             \end{aligned}\\]
                                  
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.


Overfull \hbox (69.64299pt too wide) in paragraph at lines 842--842
[]$[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (136.88995pt too wide) in paragraph at lines 842--842
[] [] 
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 842--842
\OT1/cmr/m/n/10 =
 []


Overfull \hbox (119.6678pt too wide) in paragraph at lines 842--842
[] [] 
 []


Overfull \hbox (7.7778pt too wide) in paragraph at lines 842--842
\OT1/cmr/m/n/10 =
 []


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Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.845             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.845             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.848             #
                   ### Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.848             ##
                    ## Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.848             ###
                     # Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.848             ####
                       Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no a in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no ) in font nullfont!

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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.851             $
                   \displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (27.7778pt too wide) in paragraph at lines 851--852
[]$\OT1/cmr/m/n/10 =
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OMS/cmsy/m/n/10 ^^@
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[] [][] [] \OMS/cmsy/m/n/10 ^^B
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Overfull \hbox (114.69156pt too wide) in paragraph at lines 851--852
[] [][] []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.853             $
                   \displaystyle= 1 + 2 + 2\times 0 \times \lim\_{x\to2^+}\l...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no , in font nullfont!
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Missing character: There is no o in font nullfont!
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Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
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Missing character: There is no ! in font nullfont!
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Missing character: There is no - in font nullfont!
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Missing character: There is no s in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no s in font nullfont!
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Missing character: There is no l in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.857             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.857             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
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\OT1/cmr/m/n/10 1 +
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[] [][] []$
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Overfull \hbox (15.81711pt too wide) in paragraph at lines 853--859
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Overfull \hbox (54.00414pt too wide) in paragraph at lines 853--859
[] []$
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.860             !
                   [Desmos plot of y=sin of 5 on x minus 2](gfx/sin5onxminus...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no ! in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no D in font nullfont!
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Missing character: There is no m in font nullfont!
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Missing character: There is no " in font nullfont!
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Overfull \hbox (20.0pt too wide) in paragraph at lines 860--861
[]
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.862             <
                   !-- .element: style="text-align:center" -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
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Missing character: There is no s in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no " in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 862--863
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.864             S
                   o this limit does not exist.
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no S in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.867             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.867             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.870             #
                   ### Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.870             ##
                    ## Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.870             ###
                     # Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.870             ####
                       Proof (attempt using limit laws)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no ) in font nullfont!

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[]
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[] [][] []$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.873             $
                   \displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (27.7778pt too wide) in paragraph at lines 873--874
[]$\OT1/cmr/m/n/10 =
 []


Overfull \hbox (77.74992pt too wide) in paragraph at lines 873--874
[] [][]\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \
OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (19.4445pt too wide) in paragraph at lines 873--874
\OT1/cmr/m/n/10 2)] +
 []


Overfull \hbox (67.51384pt too wide) in paragraph at lines 873--874
[] [][]\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ] +
 []


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\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^B
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Overfull \hbox (86.40262pt too wide) in paragraph at lines 873--874
[] [][] [] \OMS/cmsy/m/n/10 ^^B
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Overfull \hbox (114.69156pt too wide) in paragraph at lines 873--874
[] [][] []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.875             $
                   \displaystyle= 1 + 2 + 2\times 0 \times \lim\_{x\to2^+}\l...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no , in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.879             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.879             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.882             #
                   ### Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.882             ##
                    ## Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.882             ###
                     # Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.882             ####
                       Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no ) in font nullfont!

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[] [][] []$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.885             $
                   \displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


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OMS/cmsy/m/n/10 ^^@
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\OT1/cmr/m/n/10 2)] +
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[] [][] []$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.887             $
                   \displaystyle= 1 + 2 + 2 \times \lim\_{x\to2^+}\left[ (x-...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no , in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.891             #
                   # Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.891             ##
                     Example
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no E in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.894             #
                   ### Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.894             ##
                    ## Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.894             ###
                     # Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in horizontal mode.
l.894             ####
                       Proof (using limit laws and squeeze theorem)
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no ) in font nullfont!

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\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^B
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Overfull \hbox (30.02084pt too wide) in paragraph at lines 887--896
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[] [][] []$ 
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.897             $
                   \displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (27.7778pt too wide) in paragraph at lines 897--898
[]$\OT1/cmr/m/n/10 =
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Overfull \hbox (77.74992pt too wide) in paragraph at lines 897--898
[] [][]\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \
OMS/cmsy/m/n/10 ^^@
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\OT1/cmr/m/n/10 2)] +
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Overfull \hbox (67.51384pt too wide) in paragraph at lines 897--898
[] [][]\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ] +
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Overfull \hbox (12.77782pt too wide) in paragraph at lines 897--898
\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^B
 []


Overfull \hbox (147.07341pt too wide) in paragraph at lines 897--898
[] [][] []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.899             $
                   \displaystyle= 1 + 2 + 2 \times \lim\_{x\to2^+}\left[ (x-...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no , in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (27.7778pt too wide) in paragraph at lines 899--901
[]$\OT1/cmr/m/n/10 =
 []


Overfull \hbox (12.77782pt too wide) in paragraph at lines 899--901
\OT1/cmr/m/n/10 1 +
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\OT1/cmr/m/n/10 2 +
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Overfull \hbox (12.77782pt too wide) in paragraph at lines 899--901
\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^B
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Overfull \hbox (147.07341pt too wide) in paragraph at lines 899--901
[] [][] []$
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.902             B
                   ut
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
! Undefined control sequence.
l.903             $-1 \lt
                          \sin\left(\frac5{x-2}\right) \lt 1$, so $-(x-2) \l...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.903 ... $-1 \lt \sin\left(\frac5{x-2}\right) \lt
                                                   1$, so $-(x-2) \lt (x-2)\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
! Undefined control sequence.
l.903 ...\frac5{x-2}\right) \lt 1$, so $-(x-2) \lt
                                                   (x-2)\sin\left(\frac5{x-2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.903 ...\lt (x-2)\sin\left(\frac5{x-2}\right) \lt
                                                   (x-2)$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 902--906
[]
 []


Overfull \hbox (64.17084pt too wide) in paragraph at lines 902--906
\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1 [] [] 1$
 []


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 []


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\OT1/cmr/m/n/10 2)$
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 \OMS/cmsy/m/n/10 ^^@
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Overfull \hbox (65.84717pt too wide) in paragraph at lines 902--906
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Overfull \hbox (8.88892pt too wide) in paragraph at lines 902--906
\OT1/cmr/m/n/10 2)$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.907             H
                   ence $\displaystyle\lim\_{x\to2^+}f(x) = 1+2 + 2\times0 =...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no H in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!
! Misplaced alignment tab character &.
l.907 ...\_{x\to2^+}f(x) = 1+2 + 2\times0 = 3$.  &
                                                  nbsp; □  &nbsp; &nbsp; W...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!

! LaTeX Error: Unicode character □ (U+25A1)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.907 ...+}f(x) = 1+2 + 2\times0 = 3$.  &nbsp; □
                                                    &nbsp; &nbsp; What needs...

You may provide a definition with
\DeclareUnicodeCharacter 

! Misplaced alignment tab character &.
l.907 ...(x) = 1+2 + 2\times0 = 3$.  &nbsp; □  &
                                                  nbsp; &nbsp; What needs ch...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
! Misplaced alignment tab character &.
l.907 ...+2 + 2\times0 = 3$.  &nbsp; □  &nbsp; &
                                                  nbsp; What needs changing ...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no W in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ? in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no > in font nullfont!

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[] [][]\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =

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\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^B
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\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 !
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.914       <
             script src="/dist/reveal.js"></script>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
! Misplaced alignment tab character &.
l.920         // More info about initialization &
                                                  config:
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 8 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no O in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no B in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
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Missing character: There is no / in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no / in font nullfont!
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Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no / in font nullfont!
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Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no H in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
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Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no K in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no X in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no l in font nullfont!
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Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no d in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
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Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
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Missing character: There is no v in font nullfont!
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Missing character: There is no r in font nullfont!
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Missing character: There is no l in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no ' in font nullfont!
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Missing character: There is no p in font nullfont!
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Missing character: There is no e in font nullfont!
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Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
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Missing character: There is no n in font nullfont!
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Missing character: There is no " in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no " in font nullfont!
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Missing character: There is no ' in font nullfont!
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Missing character: There is no . in font nullfont!
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Missing character: There is no a in font nullfont!
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Missing character: There is no . in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 914--940
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.941       /
             / if the initial fragment on a slide has been defined with a 'p...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
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Missing character: There is no s in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no a in font nullfont!
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Missing character: There is no ' in font nullfont!
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Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no / in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no d in font nullfont!
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Missing character: There is no n in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no E in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
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Missing character: There is no ( in font nullfont!
Missing character: There is no ' in font nullfont!
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Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
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Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
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Missing character: There is no ' in font nullfont!
Missing character: There is no , in font nullfont!
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Missing character: There is no n in font nullfont!
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Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no ) in font nullfont!
! Misplaced alignment tab character &.
l.944 ...ent.hasAttribute('data-fragment-index') &
                                                  & event.fragment.classList...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Misplaced alignment tab character &.
l.944 ...nt.hasAttribute('data-fragment-index') &&
                                                   event.fragment.classList....
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no [ in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ' in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no R in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no > in font nullfont!
)
! Emergency stop.
<*> lectureslides/week4/week4.html
                                  
*** (job aborted, no legal \end found)

 
Here is how much of TeX's memory you used:
 34 strings out of 474222
 696 string characters out of 5748734
 1979975 words of memory out of 5000000
 22383 multiletter control sequences out of 15000+600000
 558069 words of font info for 36 fonts, out of 8000000 for 9000
 1141 hyphenation exceptions out of 8191
 23i,8n,24p,774b,107s stack positions out of 10000i,1000n,20000p,200000b,200000s
!  ==> Fatal error occurred, no output PDF file produced!