Differantiation proof question

[transgalactic]

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 got this solution :
First consider $\lim_{m\to\infty}\frac{f(2m)}{m}$, let $2m=x$ and this limit becomes $2\lim_{x\to\infty}\frac{f(x)}{x}=0$. So $\lim_{x\to\infty}\frac{f(2x)}{x}$ exists and equals $0$. So $\lim_{x\to\infty}\frac{f(x)}{x}=0\implies \lim_{x\to\infty}\frac{f(2x)-f(x)}{x}=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0$. So now consider the interval $[x,2x]$ 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

 

[rootX]

L'Hospital rule?

 

[transgalactic]

what about it?

 

[rootX]

I thought you could use L'Hospital here... using that
you get
lim .. f'(x)/1 = 0
but f(x)/x may not be in the form of inf/inf etc.

I couldn't understand the solution.