1) so rearranging gives , or sending , .
2) It comes down to showing that the limiting ratio exists. Use l'Hopitals rule. Note that the result is in agreement with (1), which is always good!
a)
Suppose f(x)=xg(x) for some function g which is continuous at 0. Prove that f is differentiable at 0 and find f'(0) in terms of g.
b)
Suppose f is differentiable at 0 and that f(0)=0, prove that f(x)=xg(x) for some function g which is continuous at 0. Hint: what happens if you try to write g(x)=f(x)/x ?