# 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