Hospital/Differentiable in inner interval/Fact

L'Hôpital's rule

Let ${}I\subseteq \mathbb {R}$ denote an open interval, and let ${}a\in I$ denote a point. Suppose that

$f,g\colon I\longrightarrow \mathbb {R}$

are continuous functions, which are differentiable on ${}I\setminus \{a\}$, fulfilling ${}f(a)=g(a)=0$, and with ${}g'(x)\neq 0$ for ${}x\neq a$. Moreover, suppose that the limit

${}w:=\operatorname {lim} _{x\rightarrow a}\,{\frac {f'(x)}{g'(x)}}\,$

exists.

Then also the limit

$\operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}$

exists, and it also equals ${}w$.

Proof, Write another proof