Find an example of a function $\displaystyle f0,1) \rightarrow \mathbb {R} $ that is differentiable with $\displaystyle f'(x)$ not bounded, and that $\displaystyle \lim _{x \rightarrow 0^+ } f(x) $ do not exist.

Umm... I can't really think of any examples. Since $\displaystyle f(x)= \frac {1}{x} $ has a bounded derivative.