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 $\left({a \,.\,.\, b}\right)$.

Suppose that $\forall x \in \left({a \,.\,.\, b}\right): g^{\prime} \left({x}\right) \ne 0$.

Suppose that $f \left({a}\right) = g \left({a}\right) = 0$.

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

provided that the second limit exists.

Proof
Take the Cauchy Mean Value Theorem with $b = x$:
 * $\exists \xi \in \left({a \,.\,.\, x}\right): \dfrac {f' \left({\xi}\right)} {g' \left({\xi}\right)} = \dfrac {f \left({x}\right) - f \left({a}\right)} {g \left({x}\right) - g \left({a}\right)}$

Then if $f \left({a}\right) = g \left({a}\right) = 0$ we have:
 * $\exists \xi \in \left({a \,.\,.\, x}\right): \dfrac {f' \left({\xi}\right)} {g' \left({\xi}\right)} = \dfrac {f \left({x}\right)} {g \left({x}\right)}$

Note that $\xi$ depends on $x$; that is $\xi$ is a function of $x$.

It follows from Limit of Function in Interval that $\xi \to a$ as $x \to a$.

Also, $\xi \ne a$ when $x > a$.

So from Hypothesis 2 of Limit of Composite Function, it follows that:
 * $\displaystyle \lim_{x \to a^+} \frac {f' \left({\xi}\right)} {g' \left({\xi}\right)} = \lim_{x \to a^+} \frac {f' \left({x}\right)} {g' \left({x}\right)}$

Hence the result.

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.