use the definitions of limit and differentiation

Mathematics
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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.

Oct 22nd, 2015

Thank you for the opportunity to help you with your question!

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

Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 22nd, 2015
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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)?

Oct 22nd, 2015

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)?

Oct 22nd, 2015

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

Oct 22nd, 2015

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Oct 22nd, 2015
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Oct 22nd, 2015
Dec 10th, 2016
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