I try to solve the following exercise:
Let $G \subset \mathbb{C}$ be a domain and $f,g$ holomorphic functions on $G \setminus\{z_0\}$ for some $z_0 \in G$. Assume that $\lim_{z\rightarrow z_o} f(z) = 0 = \lim_{z\rightarrow z_o} g(z)$ and that the limit $$c:=\lim_{z\rightarrow z_0} \frac{f'(z)}{g'(z)}$$ exists. Prove that $$\lim_{z\rightarrow z_0} \frac{f(z)}{g(z)}=c.$$
I have to use Riemann’s Theorem on removable singularities here. My attempt was to show that if $\lim_{z\rightarrow z_0} \frac{f'(z)}{g'(z)}$ exists, then there exists a neighborhood around $0$ such that $f,g$ are bounded and then I can use Riemann's Theorem to show that the functions can be extended. But I think this does not help me. Moreover I tried to use L'Hospital but I do not even know if this works in $\mathbb{C}$.
Can someone give me a hint?