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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 $$\lim_{x \to0^+} \frac{f(x)}{x}=0$$

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    You might be able to use http://math.stackexchange.com/questions/89575/how-can-i-prove-infinitesimal-limit Martin Sleziak's answer in particular.2012-01-10

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