How can I prove the limit of a quotient using the definition of derivative?

[Benzoate]

Homework Statement

Supposed f(z(0))=g(z(0))=0 and that f'(z(0)) and g'(z(0)) exist where g'(z(0)) is not equal to 0. Use definition (10, section 19 of derivative to show that :

lim z->z(0) (f(z)/g(z))=f'(z(0))/g'(z(0))

Homework Equations

definition 1: f'(z(0))=lim z->z(0) f(z)-f(z(0))/(z-z(0)

The Attempt at a Solution

f'(z(0))= lim (z->z(0)) f(z)-f(z(0))/(z-z(0))=f(z)-0/(z-z(0))=f(z)/(z-z(0)); Not sure where I should with with this proof. Help.

 

[Gib Z]

You started off really good, except you forgot the carry the limit operator through, which is important. Now that you've simplified f'(z(0)), do the same for g'(z(0)) ! Remember to use all the information you're given.