About the conditions of l'Hôpital's rule.

Question

I am reading “Calculus” (University of Tokyo Press) written by Prof. Takeshi Saito.

In this book, the conditions of l'Hôpital's rule are written as follows.

Let $f(x)$ and $g(x)$ be differentiable functions defined on some open interval $(u, a)$ and $g(x) \neq 0$ on $(u, a)$. Furthermore, let $g'(x) > 0$ on $(u, a)$ or $g'(x) < 0$ on $(u, a)$. Let the left limit $\lim_{x \to a-0} \frac{f'(x)}{g'(x)}$ converge and its value be c.

Why did he write “$g'(x) > 0$ on $(u, a)$ or $g'(x) < 0$ on $(u, a)$”? I think it is enough to write $g'(x) \neq 0$.

Answer 1

Due to Darboux theorem, if $g^\prime$ changes sign, there is a point $x$ where $g^\prime(x) = 0$. It follows that the condition $g^\prime(x)\not = 0$ on $(u, a)$ is equivalent to the book's condition.