how a sequence relates to a derivative?

i'm having some trouble with the ideas behind this question.

f is a continuous, defferentiable function with f(0)=0

given prove that

so i need a way to relate the value that some sequence converges to and the first derivative of a related function evaluated at 0.

i'm having a really hard time with this question. effectively i've got nothing on it and its been a good 5-7 hours so i don't think i can solve it with what i know already.

Re: how a sequence relates to a derivative?

This is the indeterminate form 0/0, so apply L'Hôpital's rule to get the desired result.

Re: how a sequence relates to a derivative?

so the numerator would be by the chain rule

and the denominator

so then i have

and since 1/n goes to 0 then it becomes

and i'm done?

Re: how a sequence relates to a derivative?

Your numerator is correct, but the denominator would simply be (1/n)', which cancels with this same factor in the numerator that results from the chain rule.

After that, what you wrote is correct, except let n go to infinity rather than x.

Re: how a sequence relates to a derivative?

Re: how a sequence relates to a derivative?

by the chain rule

and the denominator

so then i have

and since 1/n goes to 0 then it becomes

and i'm done?

so then this? sorry im exhausted.

Re: how a sequence relates to a derivative?