What is the value of the given limit? 
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How can I prove Infinitesimal Limit 

Let $$\lim_{x\to 0}f(x)=0$$ and $$\lim_{x\to 0}\frac{f(2x)-f(x)}{x}=0$$
Then what is the value of  $$\lim_{x\to 0}\frac{f(x)}{x}$$

If $\displaystyle\lim_{x\to0}\frac{f(x)}{x}=L$ then
$$
\lim_{x\to0}\frac{f(2x)}{x} =  2\lim_{x\to0}\frac{f(2x)}{2x} = 2\lim_{u\to0}\frac{f(u)}{u} = 2L.
$$
Then
$$
\lim_{x\to0}\frac{f(2x)-f(x)}{x} = \lim_{x\to0}\frac{f(2x)}{x} - \lim_{x\to0}\frac{f(x)}{x} =\cdots 
$$
etc.
Later note: What is written above holds in cases in which $\displaystyle\lim_{x\to0}\frac{f(x)}{x}$ exists.  The question remains: If both $\lim_{x\to0}f(x)=0$ and $\lim_{x\to0}(f(2x)-f(x))/x=0$ then does it follow that $\displaystyle\lim_{x\to0}\frac{f(x)}{x}$ exists?
