I read the following version of L'Hôpital Rule:
Let $f,g: A \subset \mathbb{R} \to \mathbb{R}$ two derivable functions in an open set, and let $a\in A$ such that one of the following conditions holds:
$$\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$$ $$f \quad \text{and} \quad g \quad \text{diverge as} \quad x\to a$$
Then, if $\lim_{x\to a} \frac{f'(x)}{g'(x)}$ exists, so does $\lim_{x\to a} \frac{f'(x)}{g'(x)}$ and they are the same.
I know that the conditions of the existence of $\lim_{x\to a} \frac{f'(x)}{g'(x)}$ means it is finite or $\pm\infty$. Previously, I studied another version which had the condition that $g'(x)\ne 0$ near $a$, but I though about it and conclude that this condition is actually assured if we have that $\lim_{x\to a} \frac{f'(x)}{g'(x)}$ exists.
I have only one last doubt that is bugging me. When the theorem says that $f$ and $g$ are divergent, is it supposed to include the case of a function that diverges but does not diverge neither to $\infty$ nor to $\infty$? Think of a function like $f(x) = x\sin{(x)}$. It diverges (the norm goes to $\infty$ as $x\to \infty$), but it does not diverge neither to $\infty$ nor to $\infty$.
Thanks in advance.