Let f: (0,1) --> R be any function with the following properties.

1) $\displaystyle \lim_{x\to 0}{f(x)} = 0$

2) $\displaystyle \lim_{x\to 0}({\frac{f(x)}{x} - \frac{f(x/2)}{x}}) = 0$

Then find: $\displaystyle \lim_{x\to 0}{\frac{f(x)}{x}$

I know that f(x) = sin(x^2) follows the above properties. Thus, $\displaystyle \lim_{x\to 0}{\frac{f(x)}{x} = 0$. But I can't prove this for any general function.

Any hintss to get me started?