Sorry, I'm not getting how you might get the answer 1 from this?
I thought you couldn't use L'Hopital's on this $\displaystyle \lim_{x \rightarrow 0^+}\frac{\ln (\ln x) }{\frac{1}{x}}$ as the numerator is undefined, and hence I don't see how you arrive at the indeterminate form $\displaystyle \frac{\infty}{\infty}$.