Limits to Infinity

We will now look at another type of limit, that is one that goes to infinity as $x$ approaches a finite value $a$. We will first look at the informal definition:

Definition (Informal): If $f$ is a function, then $\lim_{x \to a} f(x) = \infty$ if as $x$ approaches $a$, then $f(x)$ approaches $\infty$ from both the left and right sides. Similarly we say $\lim_{x \to a} f(x) = -\infty$ if as $x$ approaches $a$, then $f(x)$ approaches $-\infty$ from both the left and right sides.

The following definition is the precise, formal definition of a limit. Depending on the Calculus course you are taking, you may or may not be required to understand the following definition. If not, feel free to skip it!

Definition (Formal): Suppose that $f$ is a function defined on an open interval $I$ containing the value $a$ though not necessarily $a$. It thus follows that $\lim_{x \to a} f(x) = \infty$ means that for every number $k > 0$, there exists a $\delta > 0$ such that if $0 < \mid x - a \mid < \delta$, then $f(x) > k$. Similarly, $\lim_{x \to a} f(x) = -\infty$ means that for every number $k < 0$, there exists a $\delta > 0$ such that if $0 < \mid x - a \mid < \delta$ then $f(x) < k$.

Now we had looked at an example of an infinite earlier, $\lim_{x \to 0} \frac{1}{x^2} = \infty$. We will now look at some more examples of infinite limits.

Example 1

Evaluate the following limit $\lim_{x \to 0^+} \frac{1}{x^2 + x}$.

We note that as $x \to 0^+$, $x^2 + x$ gets arbitrarily small. Therefore, $x^2 + x \to 0$. Since the numerator is always 1 and the denominator approaches 0, $\lim_{x \to 0^+} \frac{1}{x^2 + x} = \infty$.

Example 2

Evaluate the following limit $\lim_{x \to 0} \frac{1}{\sin ^2 x}$.

We note that as $x \to 0^-$, $\sin^2 x \to 0$ which means that $\lim_{x \to 0^-} \frac{1}{\sin^2 x} = \infty$. Similarly, as $x \to 0^+$, $\sin^2 x \to 0$ as well, which means that $\lim_{x \to 0^+} \frac{1}{\sin ^2 x} = \infty$. Therefore $\lim_{x \to 0} \frac{1}{\sin ^2 x} = \infty$.