# Limits at infinity 2

## Definitions

- Intuitive definition of a limit at infinity
- Recall the functions \(f(x)=100\cdot2^{-x}\) and \(t(x)=10(1-2^{-x})\) from last time.
- Checking some values on the graphs, we see that the values approach 0 resp. 10 as \(x\) gets very large.
- We write this as \(\lim_{x\to\infty}f(x)=0\) and \(\lim_{x\to\infty}t(x)=10\).
*Intuitive definition of a limit at infinity.* Let \(f(x)\) be a function defined on some interval \((a,\infty)\). Then \(\lim_{x\to\infty}f(x)=L\) means that we can make sure that \(f(x)\) is arbitrarily close to \(L\) by taking \(x\) large enough.

- Limit at negative infinity.
- Consider now the function \(f(x)=\frac{x^2-1}{x^2+1}\).
- Checking out its graph, we see that \(f(x)\to1\) as \(x\to\infty\).
- Moreover, we can see that \(f(x)\) approaches 1 also when \(x\) gets very small.
- We write the latter as \(\lim_{x\to-\infty}f(x)=1\).
*Intuitive definition of a limit at minus infinity.* Let \(f(x)\) be a function defined on some interval \((-\infty,a)\). Then \(\lim_{x\to-\infty}f(x)=L\) means that we can make sure that \(f(x)\) gets arbitrarily close to \(L\) by taking \(x\) small enough.

## Horizontal asymptotes

- Horizontal asymptotes.
- Consider first the function \(f(x)=\tan(x)\) with domain \((-\frac{\pi}{2},\frac{\pi}{2})\).
- We have seen that \(\lim_{x\to(-\pi/2)^+}f(x)=-\infty\), and \(\lim_{x\to(\pi/2)^-}f(x)=\infty\).
- This is why the lines \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\) are
*vertical asymptotes* of the curve \(y=f(x)\).
- Consider now the inverse function \(f^{-1}(x)=\tan^{-1}(x)\).
- Correspondingly, we have \(\lim_{x\to-\infty}f^{-1}(x)=-\frac{\pi}{2}\) and \(\lim_{x\to\infty}f^{-1}(x)=\frac{\pi}{2}\).
- This is why the lines \(y=-\frac{\pi}{2}\) and \(y=\frac{\pi}{2}\) are horizontal asymptotes of the curve \(y=f^{-1}(x)\).
*Definition.* The line \(y=L\) is a *horizontal asymptote* of the curve \(y=f(x)\), if either \[
\lim_{x\to-\infty}f(x)=L\text{ or }\lim_{x\to\infty}f(x)=L.
\]
- Exercises. 2.6: 11, 12.

## Limit laws for limits at infinity

- We can extend the limit laws to limits at infinity using the following observation.
- Let \(f(x)=\frac{1}{x}\).
- Then we can see that \(\lim_{x\to-\infty}f(x)=0\) and \(\lim_{x\to\infty}f(x)=0\).
- This implies the following.
*Theorem.* (a) Let \(r>0\) be a rational number. Then we have \(\lim_{x\to\infty}x^{-r}=0\). (b) Suppose that \(x^{-r}\) is defined everywhere. Then we have \(\lim_{x\to-\infty}x^{-r}=0\).
- Exercises. 2.6: 18, 20, 24, 28