Asymptotics of $\ln T$ and $\ln( \ln T)$ I know that $\ln(\ln T)/\ln T\to0$ as $T\to \infty$. In my case, I'm evaluating 
$$\lim_{T\to\infty}\frac{(\ln \ln T)^{2/3}}{(\ln T)^\delta}$$
for some $\delta\in(0,1/6)$.
I think that if the following holds
$$\lim_{T\to\infty}\frac{\ln \ln T}{(\ln T)^\Delta}=0,$$
for any $\Delta>0$, then I can conclude that the limit I'm seeking is zero.
Although I cannot prove this general result.
Can someone help me with this?
Thanks in advance

This is indeed the case, because $\lim_{x\to\infty}\frac{\ln x}{x^\Delta}=0$ and hence we have for any sequence $(x_T)_T$ that diverges to infinity that $\lim_{T\to \infty} \ln x_T / x_T^\Delta=0$.
In particular then for $x_T=\ln T$.
