L'Hospital's Rule

# L'Hospital's Rule

We are now going to look at the famous, L'Hospital's (or L'Hôpital's) Rule, which is a method for determining the limit of a function that has an indeterminate form that we might not have previously been able to evaluate with our earlier limit rules and properties.

 Theorem 1 (L'Hospital's Rule): If $f$ and $g$ are both differentiable functions where $g'(x) \neq 0$ on an open interval $I$ that may contain $a$ but not necessarily $a$, then if $\lim_{x\rightarrow a} f(x) = 0$ and $\lim_{x\rightarrow a} g(x) = 0$ or if $\lim_{x\rightarrow a} f(x) = \pm \infty$ and $\lim_{x\rightarrow a} g(x) = \pm \infty$, then $\lim_{x\rightarrow a} \frac{f(x)}{g(x)} = \lim_{x\rightarrow a} \frac{f'(x)}{g'(x)}$.

We will now prove L'Hospital's Rule, however, we will instead jump straight into some various types of examples where we can apply it.