Infinite Limits

Quick Reference

Limit Type	Symbolic Notation	Meaning
Positive Infinite Limit	$\underset{x\to a}{lim}f(x)=+\mathrm{\infty }$	$f(x)$ increases without bound as x approaches a.
Negative Infinite Limit	$\underset{x\to a}{lim}f(x)=-\mathrm{\infty }$	$f(x)$ decreases without bound as x approaches a.
One-Sided Infinite Limit	$\underset{x\to a^{+}}{lim}f(x)=\pm \mathrm{\infty }$, $\underset{x\to a^{-}}{lim}f(x)=\pm \mathrm{\infty }$	$f(x)$ grows or decreases without bound from one side of a.

Consider the function f defined by the equation

$$f(x)=\frac{1}{(x-1{)}^2}$$

The graph of this function is illustrated below.

Note that f is not defined at $x=1$ (division by zero is not defined), but let's consider the values of f when x is close to 1. Letting x approach 1 from both sides, the corresponding values of f are given in the following table.

From this table, we see that as x gets closer and closer to 1 but never quite equal to 1, $f(x)$ ultimately becomes and remains greater than any assigned number. To express that $f(x)$ increases without bound as x approaches 1, we write

$$\underset{x\to 1}{lim}\frac{1}{(x-1{)}^2}=+\mathrm{\infty }$$

Instead of $+\mathrm{\infty }$, some books simply write $\mathrm{\infty }$.

[!NOTE]
Note that $+\mathrm{\infty }$ is not a number; it is merely a symbol indicating a mode of limiting behavior.

If $\underset{x\to a}{lim}f(x)=+\mathrm{\infty }$, we say "$f(x)$ approaches positive infinity as x approaches a," "the limit of $f(x)$, as x approaches a, is positive infinity," or "$f(x)$ increases without bound as x approaches a."

Now consider the function g defined by the equation

$$g(x)=-\frac{1}{(x-1{)}^2}.$$

The graph of $g(x)$ is shown below. As x approaches 1 from either side, $g(x)$ decreases without bound. In other words, as x approaches 1, $g(x)$ becomes and remains less than any assigned negative number. In this case, we write

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

One-sided limits can be defined accordingly. For example, consider the function $h(x)$ defined by the equation

$$h(x)=\frac{1}{x-2}.$$

The graph of this function is represented in the figure above. In this case, we write

$$\underset{x\to 2^{-}}{lim}h(x)=-\mathrm{\infty },\text{and}\underset{x\to 2^{+}}{lim}h(x)=+\mathrm{\infty }.$$