This is from baby Rudin Ch 5, exer 10:
Suppose $f$ and $g$ are complex differentiable functions on $(0,1)$, $f(x)\to0, g(x)\to0, f'(x)\to A, g'(x)\to B$ as $x\to0$, where $A$ and $B$ are complex numbers, $B\ne0$. Prove that $$\lim\limits_{x\to0}\frac{f(x)}{g(x)}=\frac{A}{B}.$$
What I can't get is how do we know that $g(x)\ne0$ in an nbd of $0$.