= Solution
Now impose the stated zero endpoint values and nonvanishing assumptions near $a$. For each $x\in(a,a+\delta)$, applying the proved identity on $[a,x]$ gives some $c_x\in(a,x)$ such that
$$
\frac{f(x)}{g(x)}=\frac{f'(c_x)}{g'(c_x)}.
$$
As $x\downarrow a$, every possible $c_x$ also tends to $a$. Explicitly, for any $\varepsilon>0$, choose a right neighborhood of $a$ where $|f'/g'-L|<\varepsilon$. If $x$ lies in it, so does $c_x$, and therefore $|f(x)/g(x)-L|<\varepsilon$. This proves the endpoint zero-over-zero form of <L'Hôpital's rule>:
$$
\boxed{\lim_{x\to a+}f(x)/g(x)=L.}
$$
No continuity of either <derivative> is required, only their ratio limit and the given nonvanishing conditions.
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