L'Hôpital's Rule/Corollary 2

Corollary to L'Hôpital's Rule
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 $f \left({x}\right) \to \infty$ and $g \left({x}\right) \to \infty$ as $x \to a^+$.

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
We have that $f \left({x}\right) \to \infty$ and $g \left({x}\right) \to \infty$ as $x \to a^+$.

Thus it follows that $\dfrac 1 {f \left({x}\right)} \to 0$ and $\dfrac 1 {g \left({x}\right)} \to 0$ as $x \to a^+$.

The result follows, after some algebra.

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

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