I am trying to work through the following problem:(adsbygoogle = window.adsbygoogle || []).push({});

if function is differentiable on an interval containing 0 except possibly at 0, and it is continous at 0, and 0= f(0)= lim f ' (x) (as x approaches 0). Prove f'(0) exists and = 0.

I thought of using the definition of a limit to get to lim [ f(x)/x] then set g(x)=x and then use L'hopital's rule. The problem is - i'm not sure I can, is it enough to show that both f, g go to zero as x goes to zero to use it??

The alternative approach i was thinking of is considering two intervals (minus delta, 0) and (0, plus delta) and then using Rolle's theorem, but the solution seems to get too complicated from then.

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# When Can i apply L'Hopital's rule?

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