there is a function f(x) which is differentiable on (a,+infinity)(adsbygoogle = window.adsbygoogle || []).push({});

suppose lim [f(x)]/x =0 as x->+infinity

prove that lim inf |f'(x)|=0 as x->+infinity ?

does this expression lim inf f'(x)=0 has to be true

if not

present a disproving example

?

i got this solution :

First consider [itex]\lim_{m\to\infty}\frac{f(2m)}{m}[/itex], let [itex]2m=x[/itex] and this limit becomes [itex]2\lim_{x\to\infty}\frac{f(x)}{x}=0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(2x)}{x}[/itex] exists and equals [itex]0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(x)}{x}=0\implies \lim_{x\to\infty}\frac{f(2x)-f(x)}{x}=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0[/itex]. So now consider the interval [itex][x,2x][/itex] and apply the MVT letting x approach infinity.

I tried to follow your logic.

and i got a few question regarding it??

http://img82.imageshack.us/my.php?image=14440292jz6.gif

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# Homework Help: Differantiation proof question

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