real analysis - How does one prove L'Hôpital's rule?

L'Hôpital's rule can be stated as follows:

Let $f, g$ be differentiable real functions defined on a deleted one-sided neighbourhood$^{(1)}$ of $a$, where $a$ can be any real number or $\pm \infty$. Suppose that both $f,g$ converge to $0$ or that both $f,g$ converge to $+\infty$ as $x \to a^{\pm}$ ($\pm$ depending on the side of the deleted neighbourhood). If

$$\frac{f'(x)}{g'(x)} \to L,$$
then
$$\frac{f(x)}{g(x)} \to L,$$

where $L$ can be any real number or $\pm \infty$.

This is an ubiquitous tool for computations of limits, and some books avoid proving it or just prove it in some special cases. Since we don't seem to have a consistent reference for its statement and proof in MathSE and it is a theorem which is often misapplied (see here for an example), it seems valuable to have a question which could serve as such a reference. This is an attempt at that.

$^{(1)}$E.g., if $a=1$, then $(1,3)$ is such a neighbourhood.