## L'Hopital's Rule

Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.

### Introduction

When we began taking limits of slopes of secant lines, $\mathrm{m}_{\mathrm{sec}}=\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{\mathrm{h}}$ as $\mathrm{h} \rightarrow 0$, we frequently encountered one difficulty: both the numerator and the denominator approached $0$. And since the denominator approached $0$, we could not apply the Main Limit Theorem. In each case, however, we managed to get past this " $0/0$ " difficulty by using algebra or geometry or trigonometry, but there was no common approach or pattern. The algebraic steps we used to evaluate $\lim \limits_{h \rightarrow 0} \frac{(2+h)^{2}-4}{h}$ seem quite different from the trigonometric steps needed for $\lim \limits_{h \rightarrow 0} \frac{\sin (2+h)-\sin (2)}{h}$.

In this section we consider a single technique, called l'Hô pital's Rule (pronounced Low-Pee-Tall), which enables us to quickly and easily evaluate limits of the form " $0/0$ " as well as several other difficult forms.