Editing L'Hôpital's rule (section)

=== 2. Differentiability of functions ===
Differentiability of functions is a requirement because if a function is not differentiable, then the derivative of the function is not guaranteed to exist at each point in <math> \mathcal{I} </math>. The fact that <math> \mathcal{I} </math> is an open interval is grandfathered in from the hypothesis of the [[Cauchy's mean value theorem]]. The notable exception of the possibility of the functions being not differentiable at <math> c </math> exists because l'Hôpital's rule only requires the derivative to exist as the function approaches <math> c </math>; the derivative does not need to be taken at <math> c </math>. 

For example, let <math> f(x) = \begin{cases} \sin x, & x\neq0 \\ 1, & x=0 \end{cases} </math> , <math> g(x)=x </math>, and <math> c = 0 </math>. In this case, <math> f(x) </math> is not differentiable at <math> c </math>. However, since <math> f(x) </math> is differentiable everywhere except <math> c </math>, then <math> \lim_{x \to c}f'(x) </math> still exists. Thus, since 

<math> \lim_{x\to c} \frac{f(x)}{g(x)} = \frac{0}{0} </math> and <math> \lim_{x\to c} \frac{f'(x)}{g'(x)}  </math> exists, l'Hôpital's rule still holds.