Limits At Infinity

Quick Reference

Direction	Symbolic Notation	Meaning
Positive Infinity	$\underset{x\to +\mathrm{\infty }}{lim}f(x)=L$	$f(x)$ approaches $L$ as x increases indefinitely.
Negative Infinity	$\underset{x\to -\mathrm{\infty }}{lim}f(x)=L$	$f(x)$ approaches $L$ as x decreases indefinitely in magnitude.

Consider the function f defined by the equation

$$f(x)=\frac{x+1}{x+2}.$$

Let's investigate the behavior of f when x is positive and becomes larger and larger. From the following table and the graph of f, we see that $f(x)$ gets closer and closer to 1 as x increases indefinitely. In this case, we say f approaches 1 (or f has limit 1) as x approaches infinity and we write

$$\underset{x\to +\mathrm{\infty }}{lim}f(x)=1.$$

Now let's investigate the behavior of f when x is negative and its magnitude becomes larger and larger. In this case, we see from the following table and the graph of f that $f(x)$ gets closer and closer to 1 too. In this case, we say f approaches 1 (or f has limit 1) as x approaches minus infinity and write

$$\underset{x\to -\mathrm{\infty }}{lim}f(x)=1.$$

[In this specific example, $\underset{x\to +\mathrm{\infty }}{lim}f(x)=\underset{x\to -\mathrm{\infty }}{lim}f(x)=1$, but in general the limits of a function as $x\to +\mathrm{\infty }$ and as $x\to -\mathrm{\infty }$ may be different.]

In general, if the graph of f gets closer and closer to the horizontal line $y=L$ as x gets larger and larger, we say the limit of f as x approaches $+\mathrm{\infty }$ is $L$, and write

$$\underset{x\to +\mathrm{\infty }}{lim}f(x)=L.$$

In a similar fashion, we can define $\underset{x\to -\mathrm{\infty }}{lim}f(x)=L$.

[!NOTE]
Instead of $+\mathrm{\infty }$ we may simply write $\mathrm{\infty }$.