Prove that if $\lim_{x\to 0} f(x) = 0$ and $\lim_{x\to 0} \frac{f(2x)-f(x)}{x}= 0$, then $\lim_{x\to 0} \frac{f(x)}{x} = 0.$
I try to solve it in this way:
$f(x)$ is infinitesimal because $\lim_{x\to 0} f(x) = 0,$
$\lim_{x\to 0} \frac{f(2x)-f(x)}{x}= 0,\Rightarrow {f(2x)-f(x)}=o({x})\Rightarrow {f(2x)}=f(x)+o({x}).$
Well
$\lim_{x\to 0} \frac{f(2x)-f(x)}{x}= \lim_{x\to 0} \frac{f(x)+o({x})}{x}= \lim_{x\to 0} \frac{f(x)}{x}+\lim_{x\to 0}\frac{o({x})}{x}=0$
$\lim_{x\to 0}\frac{o({x})}{x}=0$, of course; then $\lim_{x\to 0} \frac{f(x)}{x} = 0$