Inverse rule to the L'Hôpital's rule

If in L'Hôpital's rule we have that: $f,g :(a,b)\to \mathbb{R}$, there exist $f'(x)$, $g'(x)$, and $g'(x)\ne0$,

$$\lim_{x\to a^+} \frac{f(x)}{g(x)} = L,$$

and also $\lim_{x\to a^+} f(x)=\lim_{x\to a^+} g(x) = 0$, must it be also

$$\lim_{x\to a^+}\frac{f'(x)}{g'(x)} = L$$

or not? I think it is wrong in some cases, but I can't find an example.