(L’Hopital's Rule). Suppose I is an open interval containing a, and f, g : I → R are differentiable on I, except possibly at a. Then, if lim f(x) = 0 and lim g(x) = 0, or if lim f(x) to and lim g(x) =...

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Extracted text: (L’Hopital's Rule). Suppose I is an open interval containing a, and f, g : I → R are differentiable on I, except possibly at a. Then, if lim f(x) = 0 and lim g(x) = 0, or if lim f(x) to and lim g(x) = ±o, then f(x) f'(x) lim lim g(x) x→a g'(x)' f'(x) provided the limit lim exists. g'(x)


Extracted text: Prove the "o/0" case of l'Hopital's rule. That is, prove that if f : I → R and g : I → R are differentiable functions satisfying lim f(x) = = lim g(x) for a E I, then f' (x) gʻ(x) f(x) x→a g(x) lim = L implies lim L.