# how a sequence relates to a derivative?

• November 14th 2012, 10:30 PM
bkbowser
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 $a_{n}=nf(\frac{1}{n})$ prove that $\lim_{x\to\infty}=f^\prime(0)$

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.
• November 14th 2012, 10:45 PM
MarkFL
Re: how a sequence relates to a derivative?
$\lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{f\left (\frac{1}{n} \right)}{\frac{1}{n}}$

This is the indeterminate form 0/0, so apply L'Hôpital's rule to get the desired result.
• November 14th 2012, 11:00 PM
bkbowser
Re: how a sequence relates to a derivative?
so the numerator would be $f^\prime(1/n)*(1/n)^\prime$ by the chain rule

and the denominator $f(1/n)^\prime$

so then i have $\lim_{x\to\infty} a_n = \lim_{x\to\infty} f^\prime(1/n)$

and since 1/n goes to 0 then it becomes

$\lim_{x\to\infty} a_n = \lim_{x\to\infty} f^\prime(0)$

and i'm done?
• November 14th 2012, 11:05 PM
MarkFL
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.
• November 14th 2012, 11:07 PM
bkbowser
Re: how a sequence relates to a derivative?
kk thanks.
• November 14th 2012, 11:12 PM
bkbowser
Re: how a sequence relates to a derivative?
$f^\prime(1/n)*(1/n)^\prime$ by the chain rule

and the denominator $(1/n)^\prime$

so then i have $\lim_{n\to\infty} a_n = \lim_{n\to\infty} f^\prime(1/n)$

and since 1/n goes to 0 then it becomes

$\lim_{n\to\infty} a_n = f^\prime(0)$

and i'm done?

so then this? sorry im exhausted.
• November 14th 2012, 11:17 PM
MarkFL
Re: how a sequence relates to a derivative?
Yes, that looks good!