L'Hôpital Rulez

This is not intended for Calculus students. It treats a special case and is a curiosity.

L'Hôpital's rule can handle quotients of functions when the quotient has a limit but perhaps the functions do not. Many proofs assume the derivatives exist in a deleted neighborhood and have limits or sometimes even that they are continuous. landen got interested in the case where the derivatives exist at the point in question but perhaps no place else. Then some conditions need to be put on the functions.

Let $f(x)\,$ and $g(x)\,$ be real functions with:
$f(a)=g(a)=0\,$ for some real a.
$f'(a)\,$ exists and can be 0. $g'(a)\,$ exists and is not 0.

From the definition of derivative we have that for any $\epsilon>0\,$ there is a $\delta>0\,$ such for any $h,\ |h|<\delta$ we have:
$\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h} = \lim_{h\rightarrow 0} \frac{f(a+h)}{h}=f'(a)$
$\lim_{h\rightarrow 0} \frac{g(a+h)-g(a)}{h} = \lim_{h\rightarrow 0} \frac{g(a+h)}{h}=g'(a)$
Since all the limits exist and we are not dividing by 0 we can take the quotient:
$\lim_{h\rightarrow 0} \frac{f(a+h)}{g(a+h)}=\frac{f'(a)}{g'(a)}$ This establishes l'Hôpital's rule for these functions without assuming that the derivatives are continuous.