Question

how can you prove that if f is differentiable at x=a, then lim x goes to 0 (f(a+ch)-f(a))/h= cf'(a)

Answer 1

Write H = ch, then h = H/c

So (f(a+ch)-f(a))/h = ( f(a + H) - f(a) ) / (H / c) = c . ( f(a + H) - f(a) ) / H

The limit of this last term is just the derivative of f at a, f'(a). Hence the limit of the entire term is c.f'(a)