Limits at Infinity

Limits at Infinity#

Definitions#

Definition

The limit of $f (x)$ as $x$ approaches (positive) infinity is equal to the finite number $L$ , denoted by

lim_{x \to \infty} f (x) = L

if $f (x)$ can be made as close to $L$ as we want by taking $x$ large enough.

Definition

The limit of $f (x)$ as $x$ approaches negative infinity is equal to the finite number $M$ , denoted by

lim_{x \to - \infty} f (x) = M

if $f (x)$ can be made as close to $M$ as we want by taking $x$ to be negative and sufficiently large in absolute value.

Example 1#

A limit at infinity

The following graph illustrates a function $f (x)$ that approaches a value of 400 as $x$ increases without bound. In other words,

lim_{x \to \infty} f (x) = 400.

Graph of a function with a limit at infinity

Properties of Limits at Infinity#

Properties

All Properties of Finite Limits apply when $a$ is replaced with $\infty$ or $- \infty$ . Furthermore, the following additional properties are especially useful when evaluating limits at infinity. For all $n > 0$ , we have

$lim_{x \to \infty} \frac{1}{x^{n}} = 0$
$lim_{x \to - \infty} \frac{1}{x^{n}} = 0$ , provided that $x^{n}$ is defined for $x < 0$ .

Example 2#

Limits at infinity of $1 / x$

Notice how the graph of $y = 1 / x$ approaches the $x$ -axis as $x$ approaches positive and negative infinity. In other words,

lim_{x \to - \infty} \frac{1}{x} = 0 and lim_{x \to \infty} \frac{1}{x} = 0.

Example 3#

Limits at infinity of $1 / \sqrt{x}$

Notice how the graph of $y = 1 / \sqrt{x}$ approaches the $x$ -axis as $x$ approaches positive infinity. In other words,

lim_{x \to \infty} \frac{1}{\sqrt{x}} = 0.

Furthermore, there is no discussion of the limit as $x$ approaches negative infinity since $\sqrt{x}$ is not defined for negative values of $x$ .

$Graph of a $1/\sqrt{x}$$