Limits of real functions at infinity

## Limits of real functions at infinity
            ---
            ## 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?

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            ## 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 $1$.

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

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            ## Limits at infinity

Plot of $y = \frac1{10(x-1)} + 1 + \frac12\operatorname{sgn}(x-1)$

![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.

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            ## 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 class="color1">$\forall\\, \varepsilon \gt 0$</span><br>
              <span class="color1">$\forall\\, M \gt 0$</span>

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

<span class="color6">$\exists\\, \delta \gt 0$</span><br>
              <span class="color6">$\exists\\, R \gt 0$</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 class="color5">$0 \lt \lvert x-a \rvert \lt \delta$</span><br>
              <span class="color5">$0 \lt x-a \lt \delta$</span><br>
              <span class="color5">$0 \lt a-x \lt \delta$</span><br>
              <span class="color5">$x \gt R$</span><br>
              <span class="color5">$x \lt -R$</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 class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span><br>
              <span class="color3">$f(x) \gt M$</span><br>
              <span class="color3">$f(x) \lt -M$</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 class="color1">$\forall\\, \varepsilon \gt 0$</span>
              <span class="color6">$\exists\\, R \gt 0$</span>
              such that
              <span class="color5">$x \gt R$</span>
              $\implies$
              <span class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</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 class="color1">$\forall\\, \varepsilon \gt 0$</span><br>
              <span class="color1">$\forall\\, M \gt 0$</span>

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

<span class="color6">$\exists\\, \delta \gt 0$</span><br>
              <span class="color6">$\exists\\, R \gt 0$</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;">

<span class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span><br>
              <span class="color3">$f(x) \gt M$</span><br>
              <span class="color3">$f(x) \lt -M$</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 class="color1">$\forall\\, \varepsilon \gt 0$</span>
              <span class="color6">$\exists\\, R \gt 0$</span>
              s.t.
              <span class="color5">$x \gt R$</span>
              $\implies$
              <span class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span>
            </div>
            </div>

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

<span class="color1">$\forall\\, \varepsilon \gt 0$</span>
              <span class="color6">$\exists\\, R \gt 0$</span>
              s.t.
              <span class="color5">$x \lt -R$</span>
              $\implies$
              <span class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</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 $y = \frac1{1000(x-1)^3} + \frac{-1}2\lvert x-1 \rvert(1+2\operatorname{sgn}(x-1)) + 1$

![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 class="color1">$\forall\\, \varepsilon \gt 0$</span><br>
              <span class="color1">$\forall\\, M \gt 0$</span>

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

<span class="color6">$\exists\\, \delta \gt 0$</span><br>
              <span class="color6">$\exists\\, R \gt 0$</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;">

<span class="color3">$\lvert f(x)-\ell \rvert \lt \varepsilon$</span><br>
              <span class="color3">$f(x) \gt M$</span><br>
              <span class="color3">$f(x) \lt -M$</span>

</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 class="color1">$\forall\\, M \gt 0$</span>
              <span class="color6">$\exists\\, R \gt 0$</span>
              s.t.
              <span class="color5">$x \gt R$</span>
              $\implies$
              <span class="color3">$f(x) \gt M$</span>
            </div>
            </div>