# L'Hospital's Rule Examples for Indeterminate Products

Recall from the Indeterminate Forms page that:

If we have a limit in the form $\lim_{x\rightarrow a} f(x)g(x)$, that is as $x \to a$, $f(x) \to 0$ and $g(x) \to \pm \infty$ (or the other way around), then we have a limit of **Indeterminate Form of Type $0 \times \infty$**.

Of course, we can rewrite this function that we are taking the limit of as $f(x)g(x) = \frac{f(x)}{\frac{1}{g(x)}} = \frac{g(x)}{\frac{1}{f(x)}}$.

We will now look at some examples.

## Example 1

**Evaluate the following limit $\lim_{x \to 0} \cot 2x \cdot \sin 6x$.**

Notice that as $x \to 0$, $\cot 2x \to \pm \infty$, and $\sin 6x \to 0$. Hence we can apply L'Hospital's rule after we rewrite the limit as $\lim_{x \to 0} \frac{\sin 6x}{\tan 2x}$

(1)## Example 2

**Evaluate the limit $\lim_{x \to 1^+} \ln x \cdot \tan \left ( \frac{\pi x}{2} \right)$.**

We can apply L'Hospital's rule if we rewrite this limit to get an indeterminate quotient:

(2)Now let's use L'Hospital's rule:

(3)