I have a question about limits with factorization functions.
Prove that $\lim_{N\to\infty}\frac{\log L(\sqrt{N})}{\log > L(N)^{1/\sqrt{2}}}=1$, where $L(N)=e^{\sqrt{(\ln N)(\ln \ln N)}}$.
I have gone about this in many ways; L'Hopital's rule, changing the base, rearranging the exponents, etc, but cannot find a solution. Can you give any hints?