How can I use f'(x)= lim x->0 [f(x+h)-f(x)]/h to determine whether f(x)=x |x| is differentitable at x=O?

f '(x) = lim(h->0, ((x+h)|x + h| - x |x|)/h) at x =
0 this reduces to

f '(0) = lim(h->0 (h|h| - 0)/h) =
lim(h->0 |h|)

This limit is zero both on the - side
and + side, so the limit exists and therefor the derivative exists and it is
zero.

This is not true of f(x) = |x| because if you do the
same calculation you will get - limit is -1 and the + limit is +1 and since the
derivative is discontinuous at x = 0 the derivative does not
exist.

To repeat, f(x) = x|x| the limit is zero both on the
- side and + side, so the limit exists and therefor the derivative exists and it is zero
at x = 0.