Does this limit exist (ln)? Does this limit exist and is finite?
(It goes to one from the left 

$\lim_{x\to 1^-}{{(\ln(x) \times \ln(ln(x))}})$

I'm not sure about what the $\lim_{x\to 1^-}{{(\ln(ln(x))}})$ returns here. What do I do in this case?
$\ln(\ln(x))$ is only defined for $x&gt;1$, because if $x&lt;1$, then $\ln(x)&lt;0$ which means that $\ln(\ln(x))$ is not defined.
As noted by others if $x \to 1^{-}$ then the expression $\log\log x$ is not real and hence the limit does not exist as far as we are dealing with real functions of a real variable. On the other hand if $x \to 1^{+}$ then putting $t = \log x$ we see that $t \to 0^{+}$ and then $\log x \cdot \log \log x = t\log t \to 0$.