i got this question:(adsbygoogle = window.adsbygoogle || []).push({});

there is a function f(x) which is differentiable on (a,+infinity)

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 was present whis this solution but i didnt quite understand it.

"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."

mean value theorem says [itex]f'(c)=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0[/itex]

i dont know what the value of c??

and it doesnt prove

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

??

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: How to use mean vaule theorem here

**Physics Forums | Science Articles, Homework Help, Discussion**