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 $\forall x \in \left({a \,.\,.\, b}\right): g' \left({x}\right) \ne 0$. Suppose that $f \left({x}\right) \to \infty$ and $g \left({x}\right) \to \infty$ as $x \to a^+$.

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

provided that the second limit exists.

Proof
Let $\displaystyle \lim_{x \mathop \to a^+} \frac {f' \left({x}\right)} {g' \left({x}\right)} = L$.

Let $\left({x_n}\right)$ be a sequence such that:
 * $\quad x_n \in \left({a \,.\,.\, b}\right)$ for all $n \in \N$ and $\displaystyle \lim_{n \to \infty}x_n = a$

From Intermediate Value Theorem for Derivatives and the definition of limit of real function follows that:
 * $\displaystyle \lim_{n \to \infty}g \left({x_n}\right) = \infty$

and $\left({g(x_n)}\right)$ is strictly increasing.

Consider the range $\left[{{x_{n-1}}\,.\,.\,{x_n}}\right] \subset \left({a \,.\,.\, b}\right)$ where $n \geq 2$.

By Cauchy Mean Value Theorem, there exists $c_n \in \left({x_{n-1} \,.\,.\, x_n}\right)$ such that:
 * $\displaystyle \frac{f \left({x_n}\right) - f \left({x_{n-1}}\right)}{g \left({x_n}\right) - g \left({x_{n-1}}\right)} = \frac{f' \left({c_n}\right)}{g' \left({c_n}\right)}$

From the above and the Squeeze Theorem for Real Sequences:
 * $\displaystyle \lim_{n \to \infty} c_n = a$

and:
 * $\displaystyle \lim_{n \to \infty} \frac{f \left({x_n}\right) - f \left({x_{n-1}}\right)}{g \left({x_n}\right) - g \left({x_{n-1}}\right)} = \lim_{n \to \infty}\frac{f' \left({c_n}\right)}{g' \left({c_n}\right)} = L$

So, by Stolz-Cesàro Theorem:
 * $\displaystyle \lim_{n \mathop \to \infty} \frac {f \left({x_n}\right)} {g \left({x_n}\right)} = L$

From the definition of limit of a function deduce that:
 * $\displaystyle \lim_{x \mathop \to a^+} \frac {f \left({x}\right)} {g \left({x}\right)} = L = \lim_{x \mathop \to a^+} \frac {f' \left({x}\right)} {g' \left({x}\right)}$

Remarks

 * The proof does not actually use the assumption $\displaystyle \lim_{x \mathop \to a^+} f \left({x}\right) = \infty$.


 * Cases $x \to b^-$, $x \to \pm \infty$ and $g \left({x}\right) \to -\infty$ can be proven similarly.

However, this result was in fact discovered by.

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