4.6 Limits at infinity and asymptotes

We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function $f$ defined on an unbounded domain, we also need to know the behavior of $f$ as $x\to \text{±}\infty .$ In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function $f.$

Limits at infinity

We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs , we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

Limits at infinity and horizontal asymptotes

Recall that $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ means $f\left(x\right)$ becomes arbitrarily close to $L$ as long as $x$ is sufficiently close to $a.$ We can extend this idea to limits at infinity. For example, consider the function $f\left(x\right)=2+\frac{1}{x}.$ As can be seen graphically in [link] and numerically in [link] , as the values of $x$ get larger, the values of $f\left(x\right)$ approach $2.$ We say the limit as $x$ approaches $\infty$ of $f\left(x\right)$ is $2$ and write $\underset{x\to \infty }{\text{lim}}f\left(x\right)=2.$ Similarly, for $x<0,$ as the values $\left|x\right|$ get larger, the values of $f\left(x\right)$ approaches $2.$ We say the limit as $x$ approaches $\text{−}\infty$ of $f(x)$ is $2$ and write $\underset{x\to a}{\text{lim}}f\left(x\right)=2.$

More generally, for any function $f,$ we say the limit as $x\to \infty$ of $f\left(x\right)$ is $L$ if $f\left(x\right)$ becomes arbitrarily close to $L$ as long as $x$ is sufficiently large. In that case, we write $\underset{x\to a}{\text{lim}}f\left(x\right)=L.$ Similarly, we say the limit as $x\to \text{−}\infty$ of $f\left(x\right)$ is $L$ if $f\left(x\right)$ becomes arbitrarily close to $L$ as long as $x<0$ and $\left|x\right|$ is sufficiently large. In that case, we write $\underset{x\to \text{−}\infty }{\text{lim}}f\left(x\right)=L.$ We now look at the definition of a function having a limit at infinity.

If the values $f\left(x\right)$ are getting arbitrarily close to some finite value $L$ as $x\to \infty$ or $x\to \text{−}\infty ,$ the graph of $f$ approaches the line $y=L.$ In that case, the line $y=L$ is a horizontal asymptote of $f$ ( [link] ). For example, for the function $f\left(x\right)=\frac{1}{x},$ since $\underset{x\to \infty }{\text{lim}}f\left(x\right)=0,$ the line $y=0$ is a horizontal asymptote of $f\left(x\right)=\frac{1}{x}.$