Table of Contents

L'Hospital's Rule Examples for Indeterminate Quotients
Recall from the Indeterminate Forms page:
Sometimes we have limits in the form of $\lim_{x\rightarrow a} \frac{f(x)}{g(x)}$ where as $x \to a$, $f(x) \to 0$ and $g(x) \to 0$. Thus we have a limit of Indeterminate Form of Type $\frac{0}{0}$.
For limits in the form $\lim_{x\rightarrow a} \frac{f(x)}{g(x)}$ where as $x \to a$, $f(x) \to \pm \infty$ and $g(x) \to \pm \infty$. Thus we have a limit of Indeterminate Form of Type $\frac{\infty}{\infty}$.
Now let's try out some examples using L'Hospital's rule.
Example 1
Evaluate the following limit: $\lim_{t \to 0} \frac{e^{2t}  1}{\sin t}$.
Notice that as $t \to 0$, $e^{2t}  1 \to 0$ and $\sin t \to 0$. Thus we can apply L'Hospital's rule:
(1)Example 2
Evaluate the following limit: $\lim_{t \to 1} \frac{t^6  1}{t^5  1}$.
As $t \to 1$, $t^6  1 \to 1$ and $t^5  1 \to 1$, so we can use L'Hospital's rule:
(2)Example 3
Evaluate the following limit: $\lim_{x \to 0} \frac{x 3^x}{3^x  1}$.
As $x \to 0$, $x 3^x \to 0$, and $3^x  1 \to 0$. Let's use L'Hospital's rule:
(3)