Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval I containing a. Suppose also that g'(x)/=0 on I if x/=a. Then $ \lim_{x -> a}\frac{f(x)}{g(x)}= \lim_{x-> a}\frac{f'(x)}{g'(x)} $, if the limis on the right exists (or is positive or negative infinity).
This is Elizabeth's favorite theorem.
$ \displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\a}\frac{f'(x)}{g'(x)} $,
