L'Hôpital's rule
Differentiability of functions

Differentiability of functions is a requirement because if a function is not differentiable, then the derivative of the functions is not guaranteed to exist at each point in I {\displaystyle {\mathcal {I}}}. The fact that I {\displaystyle {\mathcal {I}}} 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 c {\displaystyle c} exists because L'Hôpital's rule only requires the derivative to exist as the function approaches c {\displaystyle c}; the derivative does not need to be taken at c {\displaystyle c}.

For example, let f (x) = { sin x, x ≠ 0 1, x = 0 {\displaystyle f(x)={\begin{cases}\sin x,&x\neq 0\\1,&x=0\end{cases}}}, g (x) = x {\displaystyle g(x)=x}, and c = 0 {\displaystyle c=0}. In this case, f (x) {\displaystyle f(x)} is not differentiable at c {\displaystyle c}. However, since f (x) {\displaystyle f(x)} is differentiable everywhere except c {\displaystyle c}, then lim x → c f ′ (x) {\displaystyle \lim _{x\to c}f'(x)} still exists. Thus, since

lim x → c f (x) g (x) = 0 0 {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {0}{0}}} and lim x → c f ′ (x) g ′ (x) {\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}} exists, L'Hôpital's rule still holds.
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