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)\neq0 $ on I if $ x\neq a $.
Then
$ \lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} $,
if the limit on the right exists (or is $ \infty $ or -$ \infty $
).
This is Elizabeth's favorite theorem.
