L'Hôpital's Rule

Theorem
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Let:
 * $\forall x \in \openint a b: \map {g'} x \ne 0$

where $g'$ denotes the derivative of $g$ $x$.

Let:
 * $\map f a = \map g a = 0$

Then:
 * $\ds \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$

provided that the second limit exists.

Also known as
Because of variants in the rendition of 's name, this proof is often seen written as:
 * L'Hospital's Rule
 * de L'Hôpital's rule

and so on.