Summary of L’Hôpital’s Rule

L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ arises.
L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
The exponential function $e^x$ grows faster than any power function $x^p$ , $p>0$ .
The logarithmic function $\ln x$ grows more slowly than any power function $x^p$ , $p>0$ .

Glossary

indeterminate forms: when evaluating a limit, the forms $0/0$ , $\infty / \infty$ , $0 \cdot \infty$ , $\infty -\infty$ , $0^0$ , $\infty^0$ , and $1^{\infty}$ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is

L’Hôpital’s rule: if $f$ and $g$ are differentiable functions over an interval $a$ , except possibly at $a$ , and $\underset{x\to a}{\lim} f(x)=0=\underset{x\to a}{\lim} g(x)$ or $\underset{x\to a}{\lim} f(x)$ and $\underset{x\to a}{\lim} g(x)$ are infinite, then $\underset{x\to a}{\lim}\dfrac{f(x)}{g(x)}=\underset{x\to a}{\lim}\dfrac{f^{\prime}(x)}{g^{\prime}(x)}$ , assuming the limit on the right exists or is $\infty$ or $−\infty$