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,

\[\lim\limits_{x\to \infty} f(x) = 400.\]

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\limits_{x \to \infty} \dfrac{1}{x^n}=0 $
$\lim\limits_{x \to -\infty} \dfrac{1}{x^n}=0$, 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,

\[\lim_{x\to -\infty} \frac{1}{x} = 0 ~~~~ \hbox{and} ~~~~~ \lim_{x\to \infty} \frac{1}{x} = 0.\]

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}$$