L'Hopital's Rule

Let $L={\lim }_{x\to a}\frac{f(x)}{g(x)}$, where $a\in \mathbb{R}\cup \{\pm \infty \}$ and $f,g$ are differentiable functions at all real numbers with $|x-a|$ sufficiently small. Suppose both $f,g\to 0$ or $f,g\to \pm \infty$ as $x\to a$. Then

$$L=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}.$$

Proof §