Prove $\lim _{x\to 0^+}\left(x^{f\left(x\right)}\right)=1$

Question

Let $f(x)$ be a function with a right derivative at $x=0$ and suppose $f(0)=0$. $$\\$$ Prove $\lim _{x\to 0^+}\left(x^{f\left(x\right)}\right)=1$

I tried to apply an exponential of a logarithm to the expression and use L'Hôpital's rule, but couldn't get further.

Any help or hints appreciated.

Answer 1

See that $x^{f(x)}=(x^x)^{\frac{f(x)}{x}}\rightarrow 1^{f'(x)}=1$.