MATH2340 Week 4

## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

| $x$  | \| | $2$  | $4$   | $6$   | $8$ | $10$ | $12$ |
            |-----|-|----|-----|-----|---|----|----|
            | $f(x)$ | \| | $13$ | $\pi$ | $\frac12$ | $1$ | $1$  | $3$  |
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x$ if $x$ is an integer, and $f(x) = t$ such that $t$ is the number of pixels used to render the letter $x$ in this font, if $x$ is not an integer.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$

Most of our functions will be defined by formulae.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

If $x$ is fixed, then $f(x)$ is fixed.

$f$ is the function, $f(x)$ means "the function $f$ evaluated at the input $x$" means "the output of $f$ which corresponds to the input $x$".
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

But different inputs could give the same output.

#### Example
            $f:\mathbb R\to\mathbb R$ defined by $f(x) = x^2$ has $f(-x)=f(x)$ for all $x$, so $f$ is not injective.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.
            
            ---
            ## Functions: injectivity and surjectivity
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            #### Examples
            Which of the following are injective, surjective, bijective on domain $\mathbb R$ and codomain $\mathbb R$?

$f(x) = x^3, \\;\\; g(x) = x^3-x, \\;\\; h(x) = \frac1{\lvert x \rvert +1}, \\;\\; p(x) = \sin(x)$,

$q(x) = 5, \\;\\; r(x) = x+\lceil x \rceil$.

(How) can you restrict their domains and codomains to make them bijective?
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $f(x) = x^3$.

Every horizontal line crosses the curve $\leq1$ time, so $f$ is injective.

Every horizontal line crosses the curve $\geq1$ time, so $f$ is surjective.

$f$ is both injective and surjective, so bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed](gfx/x3.png "Desmos plot of y=x^3")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $g(x) = x^3-x$.

Some horizontal lines cross the curve $>1$ time, so $g$ is not injective.

Every horizontal line crosses the curve $\geq1$ time, so $g$ is surjective.

$g$ is not injective, so it is not bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos plot of y=x^3-x")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $h:\mathbb R\to\mathbb R$ given by $h(x) = \frac1{\lvert x \rvert +1}$
            is neither injective nor surjective.
            But

$h:[0,\infty)\to(0,1]$ would be bijective.

$h:(-\infty,0]\to(0,1]$ would be bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.png "Desmos plot of y=1/(|x|+1)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $p:\mathbb R\to\mathbb R$ given by $p(x) = \sin(x)$
            is neither injective nor surjective.
            But

$p:[\frac{-\pi}2,\frac\pi2]\to[-1,1]$ would be bijective.

$p:[\frac{-3\pi}2,\frac{-\pi}2]\to[-1,1]$ would be bijective.

$p:[-\pi,0]\to[-1,0]$ would be surjective but not injective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=sin(x)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $q:\mathbb R\to\mathbb R$ given by $q(x) = 5$
            is neither injective nor surjective.

To make $q$ bijective, we would have to its domain and codomain to single points.
            </div>
            <div style="flex:1;margin-left:2vw">

</div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $r:\mathbb R\to\mathbb R$ given by $r(x) = x+\lceil x\rceil$
            is injective but not surjective.
            But

$r:(0,1]\to(1,2]$ would be bijective.

$r:(0,2]\to(1,2]\cup(3,4]$ is bijective.

$r:(0,2]\to(1,4]$ is injective but not surjective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilx.png "Desmos plot of y=x+ceil(x)")
            </div>
            </div>

---
            ## Limit estimation
            Let $f:\mathbb R \to\mathbb R$ and $a\in\mathbb R$.

Informally, $\displaystyle \ell=\lim_{x\to a}f(x)$ is "the value $f(x)$ gets close to when $x$ gets close to $a$".

We need a more formal definition of limits for this course, because we need to be able to *prove* what $\ell$ is.

We need to prove what $\ell$ is because estimating can result in mistakes.

[Excel file demonstrating the dangers of estimating limits](https://nc.dasmithmaths.com/index.php/s/H2HgtQMrWDxbiJB)
            
            ---
            ## Limits of real functions
            #### Definition
            Suppose $c\lt a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
            Then

$f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Or, in words,
            <span style="color:var(--uswds-red-cool-20)">No matter how small we pick $\varepsilon>0$</span>,

<span style="color:var(--uswds-indigo-20)">we can always go close enough to $a$ (ie. $\delta$ close) to ensure that</span>

#### Notation
            In this case, we write $\displaystyle \lim\_{x\to a} f(x) = \ell$.

[Excel file demonstrating how a limit is defined](https://nc.dasmithmaths.com/index.php/s/b29754YarxgnEsC)
            
            ---
            ## Limits of real functions
            #### Definition
            Suppose $c\lt a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
            Then

$f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Or, in words,
            <span style="color:var(--uswds-red-cool-20)">No matter how small we pick $\varepsilon>0$</span>,

<span style="color:var(--uswds-indigo-20)">we can always go close enough to $a$ (ie. $\delta$ close) to ensure that</span>

<span style="color:var(--uswds-green-cool-20)">all $x$ closer to $a$ (ie. within $\delta$ of $a$) give</span>
            <span style="color:var(--uswds-gold-20)">$f(x)$ within $\varepsilon$ of $\ell$</span>.
            #### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.

#### Theorem
            If $f(x)=x^2$ then $\lim_{x\to0}f(x) = 0$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\sqrt\varepsilon$.
            If $\lvert x-0 \rvert \lt \delta$, then
            $\left\lvert f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2 \lt \delta^2 = \varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=x^2$ then $\lim_{x\to0}f(x) = 0$.
            #### Rough work
            I want to make
            $\left\lvert f(x)-0 \right\rvert \lt \varepsilon$.
            I calculate $\left\lvert f(x)-0 \right\rvert = \left\lvert x^2 \right\rvert = x^2$

and I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, which implies $x^2 \lt \delta^2$.

So I just need $\delta^2 \leqslant \varepsilon$.
            So choose $\delta = \sqrt\varepsilon$.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

and I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, which implies $x^2 \lt \delta^2$.

So I just need $\delta^2 \leqslant \varepsilon$.
            So choose $\delta = \sqrt\varepsilon$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\sqrt\varepsilon$.
            If $\lvert x-0 \rvert \lt \delta$, then
            $\left\lvert f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2 \lt \delta^2 = \varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=\frac{x+1}{2-x}$ then $\lim_{x\to1}f(x) = 2$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\min\\{\frac\varepsilon6,\frac12\\}$.

If $\lvert x-1 \rvert \lt \delta$, then $6\lvert x-1 \rvert\lt\varepsilon$ and $\lvert 2-x \rvert = 2-x \gt \frac12$,

so
            $\left\lvert\frac{x+1}{2-x} - 2 \right\rvert = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt \frac{3\lvert x-1 \rvert}{\frac12} = 6 \lvert x-1 \rvert\lt\varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=\frac{x+1}{2-x}$ then $\lim_{x\to1}f(x) = 2$.
            #### Rough work
            I want to make
            $\left\lvert f(x)-2 \right\rvert \lt \varepsilon$.
            I calculate $\left\lvert f(x)-2 \right\rvert = \ldots = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}$

and I'm allowed to insist $\lvert x-1 \rvert \lt \delta$.
            If $x$ is close to $1$ then $\lvert 2-x \rvert$ is close to $1$ too.

Quantitatively, if $\lvert x-1 \rvert \lt \frac12$, then $\frac12 \lt \lvert 2-x \rvert \lt \frac32$, so $\frac1{\lvert 2-x \rvert} \lt 2$.

That tells me $\left\lvert f(x)-2 \right\rvert = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt 6\lvert x-1 \rvert$.

Now to achieve $\left\lvert f(x)-2 \right\rvert \lt \varepsilon$, all I need is $6\lvert x-1 \rvert \lt \varepsilon$; equivalently $\lvert x-1 \rvert \lt \frac\varepsilon6$.

To make all this work, I need both $\lvert x-1 \rvert \lt \frac12$ and $\lvert x-1 \rvert \lt \frac\varepsilon6$.

So choose $\delta = \min\\{\frac\varepsilon6,\frac12\\}$.
            
            ---
            ## Limits of real functions
            ### Why not input at the limit point?
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

for all $\varepsilon>0$,   $\exists \delta>0$ such that   $0 \lt \lvert x-a \rvert \lt \delta \implies$$\lvert f(x)-\ell \rvert < \varepsilon$.

Why $0 \lt \lvert x-a \rvert \lt \delta$ not just $\lvert x-a \rvert \lt \delta$?

Consider $\chi\_{\\{3\\}}:\mathbb R\to\mathbb R$, the indicator function of the set $\\{3\\}$, which is $0$ everywhere except $1$ if the input is $3$.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=indicator function for set {3} of x](gfx/chi3.png "Desmos plot of y=\chi_{3}(x)")
            </div>
            </div>

We want limits to be defined so that $\displaystyle \lim\_{x\to3}\chi\_{\\{3\\}}(x) = 0$.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Proof
            $f(2) = 5(2)+3-\chi\_{\\{2\\}}(1) = 10+3-1=12$.

Your turn!
            $\ldots$
            #### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Rough work
            If $x\approx2$ but $x\neq2$, then $f(x)\approx 5 (2) + 3 - 0 = 13$, so guess that the limit is $13$.

I want to make
            $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$.
            Suppose $x\neq2$.
            Then $f(x)=5x+3-0$.

I calculate $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-13 \right\rvert = 5\left\lvert x-2 \right\rvert$

and I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \delta$.
            Now to achieve $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$, all I need is $5\lvert x-2 \rvert \lt \varepsilon$; equivalently $\lvert x-2 \rvert \lt \frac\varepsilon5$.
            So choose $\delta = \frac\varepsilon5$.
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Rough work
            If $x\approx2$ but $x\neq2$, then $f(x)\approx 5 (2) + 3 - 0 = 13$, so guess that the limit is $13$.

I want to make
            $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$.
            Suppose $x\neq2$.
            Then $f(x)=5x+3-0$.

I calculate $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-13 \right\rvert = 5\left\lvert x-2 \right\rvert$

and I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \delta$.
            Now to achieve $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$, all I need is $5\lvert x-2 \rvert \lt \varepsilon$; equivalently $\lvert x-2 \rvert \lt \frac\varepsilon5$.
            So choose $\delta = \frac\varepsilon5$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\frac\varepsilon5$.
            If $0 \lt \lvert x-2 \rvert \lt \delta$, then $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-\chi\_{\\{2\\}}(x)-13 \right\rvert = 5 \lvert x-2 \rvert\lt5\frac\varepsilon5=\varepsilon$.   □
            
            ---
            ## Limits of real functions
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot of $y=f(x)$.

![Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgnxminus2.png "Desmos plot of y=x+sgn(x-2)")

</div>
            <div style="flex:0.3 0.3 0px;margin-left:2vw">

$\displaystyle f(2)=$

$\displaystyle\lim\_{x\to3}f(x)=$

$\displaystyle\lim\_{x\to2}f(x)=$ 
            </div>
            <div style="flex:1.7 1.7 0px;margin-left:2vw">

</div>
            </div>
            
            ---
            ## Limits of real functions
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot of $y=f(x)$.

![Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgnxminus2.png "Desmos plot of y=x+sgn(x-2)")

</div>
            <div style="flex:0.3 0.3 0px;margin-left:2vw">

$\displaystyle f(2)=$

$\displaystyle\lim\_{x\to3}f(x)=$

$\displaystyle\lim\_{x\to2}f(x)=$ 
            </div>
            <div style="flex:1.7 1.7 0px;margin-left:2vw">

$\displaystyle 2 \phantom{f(2)}$

$\displaystyle 4 \phantom{\lim\_{x\to3}}$

undefined $\displaystyle \phantom{\lim\_{x\to2}}$
            </div>
            </div>
            
            ---
            ## Limits of real functions
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot of $y=f(x)$.

![Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgnxminus2.png "Desmos plot of y=x+sgn(x-2)")

</div>
            <div style="flex:0.3 0.3 0px;margin-left:2vw">

$\displaystyle f(2)=$

$\displaystyle\lim\_{x\to3}f(x)=$

$\displaystyle\lim\_{x\to2}f(x)=$ 
            </div>
            <div style="flex:1.7 1.7 0px;margin-left:2vw">

$\displaystyle 2 \phantom{f(2)}$

$\displaystyle 4 \phantom{\lim\_{x\to3}}$

undefined $\displaystyle \phantom{\lim\_{x\to2}}$

but it "should" be $1$ or $3$, depending on which side we approach from.
            </div>
            </div>
            
            ---
            ## Limits of real functions
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot of $y=f(x)$.

![Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgnxminus2.png "Desmos plot of y=x+sgn(x-2)")

</div>
            <div style="flex:1.99 1.99 0px;margin-left:2vw">

#### Notation
              One sided limits are:

*the limit from the left* $\displaystyle \lim\_{x\to a^-} f(x)$ and

*the limit from the right* $\displaystyle \lim\_{x\to a^+} f(x)$.

#### Example
              for $f$ on the left,
              $\displaystyle \lim\_{x\to 2^-} f(x) = 1$
              and
              $\displaystyle \lim\_{x\to 2^+} f(x) = 3$.

#### Theorem
              $\displaystyle \lim\_{x\to a} f(x)$ exists if and only if both one sided limits exist and $\displaystyle \lim\_{x\to a^-} f(x) = \lim\_{x\to a^+} f(x)$.
            </div>
            <div style="flex:0.01 0.01 0px;margin-left:2vw">

</div>
            </div>
            
            ---
            ## Definitions of one sided limits
            #### Definition (recall; two sided limit)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

<span style="color:var(--uswds-red-cool-20)">for all $\varepsilon>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20);background-color:#505050">$0 \lt x-a \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$\lvert f(x)-\ell \rvert < \varepsilon$</span>.
            
            ---
            ## 📷 Algebra of limits
            If $\displaystyle\lim\_{x\to a}f(x)$ and $\displaystyle\lim\_{x\to a}g(x)$ both exist and $c\in\mathbb R$, then
            1. $\displaystyle\lim\_{x\to a}(f(x) \pm g(x)) = \left(\displaystyle\lim\_{x\to a}f(x)\right) \pm \left(\displaystyle\lim\_{x\to a}g(x)\right)$.<br><br>
            1. $\displaystyle\lim\_{x\to a}(f(x) g(x)) = \left(\displaystyle\lim\_{x\to a}f(x)\right) \left(\displaystyle\lim\_{x\to a}g(x)\right)$.<br><br>
            1. $\displaystyle\lim\_{x\to a}\left(\frac{f(x)}{g(x)}\right) = \frac{\displaystyle\lim\_{x\to a}f(x)}{\displaystyle\lim\_{x\to a}g(x)}$, provided $\displaystyle\lim\_{x\to a}g(x)\neq0$.<br><br>
            1. $\displaystyle\lim\_{x\to a}cf(x)) = c\left(\displaystyle\lim\_{x\to a}f(x)\right)$.

In each of these, $x\to a$ can be replaced by $x\to a^\pm$.
            
            ---
            ## 📷 Squeeze theorem
            Suppose $f,g,h$ are functions defined on an open interval $I$, except possibly at $a\in I$, and, for all $x\in I$, $f(x) \leqslant g(x) \leqslant h(x)$.

If $\displaystyle\lim\_{x\to a}f(x) = \ell = \displaystyle\lim\_{x\to a}h(x)$,

then also $\displaystyle\lim\_{x\to a}g(x)=\ell$.

There are also versions for one sided limits.
            
            ---
            ## Example
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0.8 0.8 0px;margin-right:2vw;background-color:white;color:black">

Plot $y=f(x)$

![Desmos plot of y=x plus sgn of x minus 2 plus 2x minus 4 times sin of 5 on x minus 2](gfx/xplussgnxminus2plus2xminus4sin5onxminus2.png "Desmos plot of y=x+sgn(x-2)+2(x-2)sin(5/(x-2))")

</div>
            <div style="flex:0.15 0.15 0px;margin-left:2vw">

$\displaystyle \phantom{\left(\frac52\right)}f(x) =$

$\displaystyle\lim\_{x\to2^-}f(x)=$

$\displaystyle\lim\_{x\to2^+}f(x)=$

$\displaystyle\lim\_{x\to2}f(x)=$ 
            </div>
            <div style="flex:2 2 0px;margin-left:2vw">

$\displaystyle H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$

$\displaystyle ? \phantom{\lim\_{2^-}}$

$\displaystyle ? \phantom{\lim\_{2^+}}$

$\displaystyle ? \phantom{\lim\_{2}}$
            </div>
            </div>
            
            ---
            ## Example
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0.8 0.8 0px;margin-right:2vw;background-color:white;color:black">

Plot $y=f(x)$

![Desmos plot of y=x plus sgn of x minus 2 plus 2x minus 4 times sin of 5 on x minus 2](gfx/xplussgnxminus2plus2xminus4sin5onxminus2.png "Desmos plot of y=x+sgn(x-2)+2(x-2)sin(5/(x-2))")

</div>
            <div style="flex:0.15 0.15 0px;margin-left:2vw">

$\displaystyle \phantom{\left(\frac52\right)}f(x) =$

$\displaystyle\lim\_{x\to2^-}f(x)=$

$\displaystyle\lim\_{x\to2^+}f(x)=$

$\displaystyle\lim\_{x\to2}f(x)=$ 
            </div>
            <div style="flex:2 2 0px;margin-left:2vw">

$\displaystyle H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$

$\displaystyle 1 \phantom{\lim\_{2^-}}$

$\displaystyle 3 \phantom{\lim\_{2^+}}$

undefined $\displaystyle \phantom{\lim\_{2}}$
            </div>
            </div>
            
            ---
            ## Example
            Let $\displaystyle f(x)= H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$.
            Then $\displaystyle\lim\_{x\to2^+}f(x)=3$.
            #### Proof (direct from definition)
            For any $\varepsilon\gt0$, let $\delta=\frac\varepsilon3$, and suppose $0 \lt x-2 \lt \delta$.

Then $x \gt 2$, so $H(x-2)=1$, so
            \\[\begin{aligned}
              \left\lvert f(x) - 3 \right\rvert
              &= \left\lvert 1+x+2(x-2)\sin\left(\frac5{x-2}\right) - 3 \right\rvert \\\\
              &= \left\lvert x+2(x-2)\sin\left(\frac5{x-2}\right) - 2 \right\rvert \\\\
              &= (x-2) \left\lvert 1 + 2\sin\left(\frac5{x-2}\right) \right\rvert
              \leqslant (x-2) 3 \\\\
              &\lt 3 \delta = 3\frac\varepsilon3=\varepsilon.
              \qquad □
            \end{aligned}\\]
            
            ---
            ## Example
            Let $\displaystyle f(x)= H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$.
            Then $\displaystyle\lim\_{x\to2^+}f(x)=3$.
            #### Proof (attempt using limit laws)
            $\displaystyle\lim\_{x\to2^+}f(x) = \lim\_{x\to2^+}\left[ H(x-2)+x+2(x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[x] + 2 \times \lim\_{x\to2^+}\left[ x-2 \right] \times \lim\_{x\to2^+}\left[ \sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= 1 + 2 + 2\times 0 \times \lim\_{x\to2^+}\left[ \sin\left(\frac5{x-2}\right) \right]$,
            provided the last limit exists.
            
            ---
            ## Example
            Plot of $\displaystyle y=\sin\left(\frac5{x-2}\right)$.

![Desmos plot of y=sin of 5 on x minus 2](gfx/sin5onxminus2.png "Desmos plot of y=sin(5/(x-2))")

So this limit does not exist.
            
            ---
            ## Example
            Let $\displaystyle f(x)= H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$.
            Then $\displaystyle\lim\_{x\to2^+}f(x)=3$.
            #### Proof (attempt using limit laws)
            $\displaystyle\lim\_{x\to2^+}f(x) = \lim\_{x\to2^+}\left[ H(x-2)+x+2(x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[x] + 2 \times \lim\_{x\to2^+}\left[ x-2 \right] \times \lim\_{x\to2^+}\left[ \sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= 1 + 2 + 2\times 0 \times \lim\_{x\to2^+}\left[ \sin\left(\frac5{x-2}\right) \right]$,
            but this did not work.
            
            ---
            ## Example
            Let $\displaystyle f(x)= H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$.
            Then $\displaystyle\lim\_{x\to2^+}f(x)=3$.
            #### Proof (using limit laws and squeeze theorem)
            $\displaystyle\lim\_{x\to2^+}f(x) = \lim\_{x\to2^+}\left[ H(x-2)+x+2(x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[x] + 2 \times \lim\_{x\to2^+}\left[ (x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= 1 + 2 + 2 \times \lim\_{x\to2^+}\left[ (x-2)\sin\left(\frac5{x-2}\right) \right]$,
            provided the last limit exists.
            
            ---
            ## Example
            Let $\displaystyle f(x)= H(x-2) + x + 2(x-2)\sin\left(\frac5{x-2}\right)$.
            Then $\displaystyle\lim\_{x\to2^+}f(x)=3$.
            #### Proof (using limit laws and squeeze theorem)
            $\displaystyle\lim\_{x\to2^+}f(x) = \lim\_{x\to2^+}\left[ H(x-2)+x+2(x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= \lim\_{x\to2^+}[H(x-2)] + \lim\_{x\to2^+}[x] + 2 \times \lim\_{x\to2^+}\left[ (x-2)\sin\left(\frac5{x-2}\right) \right]$

$\displaystyle= 1 + 2 + 2 \times \lim\_{x\to2^+}\left[ (x-2)\sin\left(\frac5{x-2}\right) \right]$,
            provided the last limit exists.

But
            $-1 \lt \sin\left(\frac5{x-2}\right) \lt 1$, so $-(x-2) \lt (x-2)\sin\left(\frac5{x-2}\right) \lt (x-2)$,
            and $\lim\_{x\to2^+} [-(x-2)] = 0 = \lim\_{x\to2^+} (x-2)$,
            so the last limit does exist, and is zero.

Hence $\displaystyle\lim\_{x\to2^+}f(x) = 1+2 + 2\times0 = 3$.    □      What needs changing for $x\to2^-$?
            
            ---
            ## Infinite limits
            #### Definition (recall; finite two sided limit)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

<span style="color:var(--uswds-red-cool-20);background-color:#505050">for all $\varepsilon>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20);background-color:#505050">$\lvert f(x)-\ell \rvert < \varepsilon$</span>.
            #### Definition (limit $\infty$)
            $f(x)$ *converges* to $\infty$ as $x\to a$ if

Complete this statement!
            
            ---
            ## Infinite limits
            #### Definition (recall; finite two sided limit)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

<span style="color:var(--uswds-red-cool-20);background-color:#505050">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20);background-color:#505050">$f(x) < -M$</span>.
            
            ---
            ## Infinite limits
            #### Definition (recall; finite two sided limit)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

<span style="color:var(--uswds-red-cool-20);background-color:#505050">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20);background-color:#505050">$f(x) < -M$</span>.
            #### Exercise
            Write down the definitions of one sided infinite limts (there are four).
            
            ---
            ## Infinite limits
            ### Reminder of some related definitions
            #### Definition (recall; limit from the right)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a^+$ if

<span style="color:var(--uswds-red-cool-20)">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$f(x) > M$</span>.
            
            ---
            ## Infinite limits
            ### Reminder of some related definitions
            #### Definition (recall; limit from the right)
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a^+$ if

<span style="color:var(--uswds-red-cool-20)">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$f(x) > M$</span>.
            #### Solution: Definition (limit $\infty$ from the right)
            $f(x)$ *converges* to $\infty$ as $x\to a^+$ if

<span style="color:var(--uswds-red-cool-20)">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt x-a \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$f(x) > M$</span>.
            
            ---
            ## Example
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot $y=f(x)$

![Desmos plot of y=1 on 1000 x minus 1 cubed](gfx/1on1000xminus1cubed.png "Desmos plot of y=1/[1000(x-1)^3]")

</div>
            <div style="flex:4.5 4.5 0px;margin-left:2vw">

$\displaystyle f(x) = \frac1{1000(x-1)^3}$.
              #### Theorem
              $\displaystyle\lim\_{x\to1^-} f(x) = -\infty$ and $\displaystyle\lim\_{x\to1^+} f(x) = \infty$, so $\displaystyle\lim\_{x\to1} f(x)$ is undefined.

#### Proof
              Try it!

#### Definition (limit $\infty$ from the right)
              $f(x)$ *converges* to $\infty$ as $x\to a^+$ if
              <span style="color:var(--uswds-red-cool-20)">for all $M>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt x-a \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$f(x) > M$</span>.
              #### How to prove convergence if you know the limit
              Construct a machine that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $M$</span>, so $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.
            </div>
            </div>
            
            ---
            ## Example
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1 1 0px;margin-right:2vw;background-color:white;color:black">

Plot $y=f(x)$

![Desmos plot of y=1 on 1000 x minus 1 cubed](gfx/1on1000xminus1cubed.png "Desmos plot of y=1/[1000(x-1)^3]")

</div>
            <div style="flex:4.5 4.5 0px;margin-left:2vw">

#### Proof
              For any $M>0$, let $\delta=\frac1{10\sqrt[3]M}$.

If $0<x-1<\delta$, then $\frac1{x-1} \gt 10 \sqrt[3]M$, so $\frac1{10(x-1)} \gt \sqrt[3]M$, so $f(x) = \left(\frac1{10(x-1)}\right)^3 \gt M$.
              Therefore $\displaystyle\lim\_{x\to1^+} f(x) = \infty$.

If instead $0<1-x<\delta$, then $\frac1{x-1} \lt - 10 \sqrt[3]M$, so $\frac1{10(x-1)} \lt -\sqrt[3]M$, so $f(x) = \left(\frac1{10(x-1)}\right)^3 \lt -M$.
              Hence $\displaystyle\lim\_{x\to1^-} f(x) = -\infty$.
            </div>
            </div>
            
            ---
            ## Limits at infinity

Plot of $y = \frac1{10(x-1)} + 1$

![Desmos plot of y=1 on 10 x minus 1 plus 1](gfx/1on10xminus1plus1.png "Desmos plot of y=1/[10(x-1)]+1")

What is the orange dashed line called?

What does it mean?

---
            ## Limits at infinity

Plot of $y = \frac1{10(x-1)} + 1$

![Desmos plot of y=1 on 10 x minus 1 plus 1](gfx/1on10xminus1plus1.png "Desmos plot of y=1/[10(x-1)]+1")

The orange line is an *asymptote*.

It shows that as $x$ gets very large (positive or negative), $f(x)$ gets close to $1$.

---
            ## Limits at infinity

Plot of $y = \frac1{10(x-1)} + 1$

![Desmos plot of y=1 on 10 x minus 1 plus 1](gfx/1on10xminus1plus1.png "Desmos plot of y=1/[10(x-1)]+1")

We say $\displaystyle \lim\_{x\to\infty} f(x) = 1$ and $\displaystyle \lim\_{x\to-\infty} f(x) = 1$.

---
            ## Limits at infinity

Plot of $y = \frac1{10(x-1)} + 1 + \frac12H(x-1)$

![Desmos plot of y=1 on 10 x minus 1 plus 1](gfx/1on10xminus1plus1plushalfsgnxminus1.png "Desmos plot of y=1/[10(x-1)]+1")

We say $\displaystyle \lim\_{x\to\infty} f(x) = 1$ and $\displaystyle \lim\_{x\to-\infty} f(x) = 0$.

They do not have to be equal.

---
            ## Limits at infinity

#### Definition
            Suppose $f:\mathbb R\to\mathbb R$ and $\ell\in\mathbb R$.

We say $\displaystyle \lim\_{x\to \infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

</div>
            </div>

We say $\displaystyle \lim\_{x\to-\infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

</div>
            </div>

#### Exercise
            Complete the definitions from:
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0.5;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-red-cool-20)">$\forall\\, \varepsilon \gt 0$</span><br>
              <span style="color:var(--uswds-red-cool-20)">$\forall\\, M \gt 0$</span>

</div>
            <div style="flex:0.5;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-indigo-20)">$\exists\\, \delta \gt 0$</span><br>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

such that

</div>
            <div style="flex:1;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta$</span><br>
              <span style="color:var(--uswds-green-cool-20)">$0 \lt x-a \lt \delta$</span><br>
              <span style="color:var(--uswds-green-cool-20)">$0 \lt a-x \lt \delta$</span><br>
              <span style="color:var(--uswds-green-cool-20)">$x \gt R$</span><br>
              <span style="color:var(--uswds-green-cool-20)">$x \lt -R$</span>

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

$\implies$

</div>
            <div style="flex:1;margin-left:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-gold-20)">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span><br>
              <span style="color:var(--uswds-gold-20)">$f(x) \gt M$</span><br>
              <span style="color:var(--uswds-gold-20)">$f(x) \lt -M$</span>

</div>
            </div>
            
            ---
            ## Limits at infinity

#### Definition
            Suppose $f:\mathbb R\to\mathbb R$ and $\ell\in\mathbb R$.

We say $\displaystyle \lim\_{x\to \infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

<span style="color:var(--uswds-red-cool-20)">$\forall\\, \varepsilon \gt 0$</span>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>
              such that
              <span style="color:var(--uswds-green-cool-20)">$x \gt R$</span>
              $\implies$
              <span style="color:var(--uswds-gold-20)">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span>
            </div>
            </div>

We say $\displaystyle \lim\_{x\to-\infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

</div>
            </div>

<span style="color:var(--uswds-red-cool-20)">$\forall\\, \varepsilon \gt 0$</span><br>
              <span style="color:var(--uswds-red-cool-20)">$\forall\\, M \gt 0$</span>

</div>
            <div style="flex:0.5;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-indigo-20)">$\exists\\, \delta \gt 0$</span><br>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

such that

</div>
            <div style="flex:1;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

$\implies$

</div>
            <div style="flex:1;margin-left:1vw;display:flex;justify-content:center;align-items:center;">

</div>
            </div>
            
            ---
            ## Limits at infinity

#### Definition
            Suppose $f:\mathbb R\to\mathbb R$ and $\ell\in\mathbb R$.

We say $\displaystyle \lim\_{x\to \infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

We say $\displaystyle \lim\_{x\to-\infty} f(x) = \ell$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

<span style="color:var(--uswds-red-cool-20)">$\forall\\, \varepsilon \gt 0$</span>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>
              such that
              <span style="color:var(--uswds-green-cool-20)">$x \lt -R$</span>
              $\implies$
              <span style="color:var(--uswds-gold-20)">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span>
            </div>
            </div>

---
            ## Infinite limits at infinity

Plot of $y = \frac1{1000(x-1)^3} + \frac12\lvert x-1 \rvert + 1$

![Desmos plot of y=1 on 1000 x minus 1 cubed plus half abs x minus 1 plus 1](gfx/1on1000xminus1cubedplushalfabsxminus1plus1.png "Desmos plot of y=1/[1000(x-1)^3]+|x-1|/2+1")

What are the orange dashed lines called?

What do they mean?

---
            ## Infinite limits at infinity

Plot of $y = \frac1{1000(x-1)^3} + \frac12\lvert x-1 \rvert + 1$

![Desmos plot of y=1 on 1000 x minus 1 cubed plus half abs x minus 1 plus 1](gfx/1on1000xminus1cubedplushalfabsxminus1plus1.png "Desmos plot of y=1/[1000(x-1)^3]+|x-1|/2+1")

The orange lines are sloped asymptotes.

They shows that as $x$ gets very large (positive or negative), $f(x)$ gets very large and positive.

---
            ## Infinite limits at infinity

Plot of $y = \frac1{1000(x-1)^3} + \frac12\lvert x-1 \rvert + 1$

![Desmos plot of y=1 on 1000 x minus 1 cubed plus half abs x minus 1 plus 1](gfx/1on1000xminus1cubedplushalfabsxminus1plus1.png "Desmos plot of y=1/[1000(x-1)^3]+|x-1|/2+1")

We say $\displaystyle \lim\_{x\to\infty} f(x) = \infty$ and $\displaystyle \lim\_{x\to-\infty} f(x) = \infty$.

---
            ## Infinite limits at infinity

Plot of $y = \frac1{1000(x-1)^3} + \frac{-1}2\lvert x-1 \rvert(1+2H(x-1)) + 1$

![Desmos plot of y=1 on 1000 x minus 1 cubed minus half abs x minus 1 times 1 plus twice signum x minus 1 plus 1](gfx/1on1000xminus1cubedminushalfabsxminus1times1plus2timessignumxminus1plus1.png "Desmos plot of y=1/[1000(x-1)^3]-|x-1|(1+sgn(x-1))/2+1")

We say $\displaystyle \lim\_{x\to\infty} f(x) = -\infty$ and $\displaystyle \lim\_{x\to-\infty} f(x) = \infty$.

They need not be the same.

---
            ## Infinite limits at infinity

#### Definition
            Suppose $f:\mathbb R\to\mathbb R$ and $\ell\in\mathbb R$.

We say $\displaystyle \lim\_{x\to \infty} f(x) = \infty$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

</div>
            </div>

We say $\displaystyle \lim\_{x\to \infty} f(x) = -\infty$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

</div>
            </div>

<span style="color:var(--uswds-red-cool-20)">$\forall\\, \varepsilon \gt 0$</span><br>
              <span style="color:var(--uswds-red-cool-20)">$\forall\\, M \gt 0$</span>

</div>
            <div style="flex:0.5;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

<span style="color:var(--uswds-indigo-20)">$\exists\\, \delta \gt 0$</span><br>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

such that

</div>
            <div style="flex:1;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

</div>
            <div style="flex:0;margin-left:1vw;margin-right:1vw;display:flex;justify-content:center;align-items:center;">

$\implies$

</div>
            <div style="flex:1;margin-left:1vw;display:flex;justify-content:center;align-items:center;">

</div>
            </div>
            
            ---
            ## Infinite limits at infinity

#### Definition
            Suppose $f:\mathbb R\to\mathbb R$ and $\ell\in\mathbb R$.

We say $\displaystyle \lim\_{x\to \infty} f(x) = \infty$ if…
            </div>
            <div style="flex:1;margin-left:1vw">

<span style="color:var(--uswds-red-cool-20)">$\forall\\, M \gt 0$</span>
              <span style="color:var(--uswds-indigo-20)">$\exists\\, R \gt 0$</span>
              such that
              <span style="color:var(--uswds-green-cool-20)">$x \gt R$</span>
              $\implies$
              <span style="color:var(--uswds-gold-20)">$f(x) \gt M$</span>
            </div>
            </div>