Why L'hopital's rule only proved for indetermined forms? In this proof of L'hopital's rule, $\lim\limits_{x\to a}\, f(x)=0$ and $g(x)=0$ seems have no role to paly. So what goes wrong when the limits mentioned before are not equal to $0$? (My guess is that you can't assume that "f(a)=g(a)=$0$" without making the functions discontinuous when those limit are not equal to $0$. Is that right?) thanks!

The proof requires an application of Rolle's theorem to

$$h(x) = f(x) - \frac{f(b)}{g(b)}g(x).$$

Since $h(b) = 0$, this requires $h(a) = 0$ for all $b >a$. This is satisfied if $f(a) = g(a) = 0$ either outright or by continuous extension using $f(x),g(x) \to 0$ as $x \to a+$.