L'Hôpital's Rule

Theorem
Let $f$ and $g$ be real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let:
 * $\forall x \in \left({a \,.\,.\, b}\right): g' \left({x}\right) \ne 0$

Let:
 * $f \left({a}\right) = g \left({a}\right) = 0$

Then:
 * $\displaystyle \lim_{x \to a^+} \frac {f \left({x}\right)} {g \left({x}\right)} = \lim_{x \to a^+} \frac {f' \left({x}\right)} {g' \left({x}\right)}$

provided that the second limit exists.

Proof 2
However, this result was in fact discovered by Johann Bernoulli.

Because of variants in the rendition of his name, this proof is often seen written as L'Hospital's Rule.