How do you prove that lim of (f(c+g(h))-f(c))/h as h goes to 0 exist? Knowing that f is differentiable at c, and g is differentiable at 0, g(0)=0, and g(x) does not equal to 0 if x is not 0. Only use the definitions of limit and differentiation.

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lim [f(c+g(h))-f(c)]/h = lim [f(c+g(h))-f(c)]/g(h) *[g(h)-g(0)]/(h-0)

=f'(c) *g'(0)

thus the limit exist

lim [f(c+g(h))-f(c)]/g(h) still has h goes to 0, not g(h) goes to 0. Is it still equal to f'(c)?

in lim f(c+g(h))-f(c))/g(h), it is g(h) --->0, because g(h) -->0 as h -->0

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