I would like to solve this problem, but I do not know how ...
Let $f:(0;1) \rightarrow \mathbb{R}$ be a function such that: $$\lim_{x \to0^+}f(x)=0$$ and such that there exists $0<\lambda<1$ such that: $$\lim_{x \to0^+} \frac{ \left [ f(x)-f(\lambda x) \right ]}{x}=0$$ prove that $$\lim_{x \to0^+} \frac{f(x)}{x}=0$$