L'Hospitals Rule on infinite intervals It seems like every proof I see of L'Hospital's rule is on a finite interval, for example my Real Analysis defines L'Hospital's rule as follows:

If $f$ and $g$ are differentiable functions defined on an interval $(a,b)$, both of which tend to $0$ at $b$, and if the ratio of their derivatives $\frac{f'(x)}{g'(x)}$ tends to a finite limit $L$ at $b$ then $\frac{f(x)}{g(x)}$ also tends to $L$ at $b$. It is assumed that $g(x), g'(x) \neq 0$

How would I go about proving the result on the interval $[a,\infty]$? What about if $L = \infty$?

For $L \to \infty$ apply the theorem to $$\lim_{x \to 0^+} \frac{F(x)}{G(x)},$$ where $F(x) = f(1/x)$ and $G(x) = g(1/x)$
