Infinite Limits

Concept	Quick Reference
$\underset{x\to a}{lim}f(x)=+\mathrm{\infty }$	$f(x)$ increases without bound as $x\to a$
$\underset{x\to a}{lim}f(x)=-\mathrm{\infty }$	$f(x)$ decreases without bound as $x\to a$
$\pm \mathrm{\infty }$ is not a number	it is a symbol describing a mode of behavior
One-sided versions	$\underset{x\to a^{-}}{lim}$ and $\underset{x\to a^{+}}{lim}$ apply as well

Limit Equal to $+\mathrm{\infty }$

Consider the function

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

The graph of this function is shown below.

The function is not defined at $x=1$. Letting $x$ approach $1$ from both sides, the values are shown in the following table.

As $x$ gets closer and closer to $1$ (from either side), $f(x)$ ultimately becomes and remains greater than any assigned number. To express that $f(x)$ increases without bound as $x\to 1$, we write:

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

Some books write $\mathrm{\infty }$ instead of $+\mathrm{\infty }$.

• 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 $a$," or "$f(x)$ increases without bound as $x$ approaches $a$."

Limit Equal to $-\mathrm{\infty }$

Now consider

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

The graph of $g(x)$ is shown below.

As $x$ approaches $1$ from either side, $g(x)$ decreases without bound, it becomes and remains less than any assigned negative number. We write:

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

One-Sided Infinite Limits

One-sided limits can be infinite as well. Consider

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

The graph is shown below.

As $x$ approaches $2$ from the left, the denominator $x-2$ is small and negative, so $h(x)\to -\mathrm{\infty }$. As $x$ approaches $2$ from the right, the denominator is small and positive, so $h(x)\to +\mathrm{\infty }$. We write:

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

Frequently Asked Questions

Is a limit of $\pm \mathrm{\infty }$ the same as saying "the limit does not exist"?

Technically an infinite limit is a special kind of non-existence, there is no finite number $L$ that $f(x)$ approaches. However, writing $\underset{x\to a}{lim}f(x)=+\mathrm{\infty }$ conveys more information than simply saying "the limit does not exist": it tells you specifically that $f(x)$ grows without bound. Many textbooks use the notation $=+\mathrm{\infty }$ to describe this behavior precisely.

How do I tell whether a one-sided infinite limit is $+\mathrm{\infty }$ or $-\mathrm{\infty }$?

Look at the sign of the fraction near the point. If the numerator has a positive limit and the denominator approaches $0$ through positive values, the result is $+\mathrm{\infty }$. If the denominator approaches $0$ through negative values and the numerator is positive, the result is $-\mathrm{\infty }$. The rule: same signs give $+\mathrm{\infty }$; opposite signs give $-\mathrm{\infty }$.

What is the connection between infinite limits and vertical asymptotes?

The line $x=a$ is a **vertical asymptote** of $f$ if any one of the one-sided limits $\underset{x\to a^{\pm }}{lim}f(x)$ equals $+\mathrm{\infty }$ or $-\mathrm{\infty }$. In other words, an infinite limit at $x=a$ is precisely the condition that produces a vertical asymptote there.