L'Hôpital's rule
Proof

Consider the functions h (x) = f (x) − f (a) {\displaystyle h(x)=f(x)-f(a)} and g (x) = x − a {\displaystyle g(x)=x-a}. The continuity of f at a tells us that lim x → a h (x) = 0 {\displaystyle \lim _{x\to a}h(x)=0}. Moreover, lim x → a g (x) = 0 {\displaystyle \lim _{x\to a}g(x)=0} since a polynomial function is always continuous everywhere. Applying L'Hopital's rule shows that f ′ (a):= lim x → a f (x) − f (a) x − a = lim x → a h ′ (x) g ′ (x) = lim x → a f ′ (x) {\displaystyle f'(a):=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}=\lim _{x\to a}{\frac {h'(x)}{g'(x)}}=\lim _{x\to a}f'(x)}.
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