The derivation of L'Hôpital's rule requires Cauchy's theorem, which, in turn, requires the following conditions:
two functions $f(x)$ and $g(x)$ must be continuous and differentiable in an interval $[a,b]$, and $g'(x)\neq 0$
If we have an indetermination for the limit
$$\lim_{x\to a}\frac{f(x)}{g(x)} $$
but, after applying L'Hôpital's rule, we get the indetermination
$$\lim_{x\to a}\frac{f'(x)}{g'(x)} = \frac{f'(a)}{g'(a)} = \frac{0}{0} $$
why can we apply L'Hôpital's rule again to $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ to solve the indetermination $\frac{0}{0}$, if the condition of Cauchy's theorem that $g'(x)\neq 0$ is broken?