Let $f:A\rightarrow \mathbb{R} $ be a function, where $A\subset \mathbb{R}$, and $x_0$ a limit point of $A$ in which $f$ is continuous, with $y_0=f(x_0)$. Suppose moreover that $g:\mathrm {Im}f\rightarrow \Bbb R$ is such that $g(y_0)=0$ and $$\lim_{y\rightarrow y_0} \frac{g(y)}{y-y_0}=0$$.
Now, is it true that $$\lim_{x\rightarrow x_0}\frac{g(f(x))}{x-x_0}=0$$?
If it is, how do you prove it rigorously?