I am trying to work through the following problem: 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.