Argue whether $\frac{1}{x}\ln(\ln(x))=o(\ln(x)/x)$ as $x\to\infty$ (or not).

Question

I am just wondering whether $$ \frac{1}{x}\ln(\ln(x))=o\left(\frac{\ln(x)}{x}\right)\text{ as }x\to\infty. $$

This is little-o-notation, meaning I have to show that for every positive constant $\varepsilon$ there exists a constant $N$ such that $$ \left\lvert\frac{\ln(\ln(x))}{x}\right\rvert\leq \varepsilon\frac{\ln(x)}{x}\text{ for all }x\geq N. $$

I think this is a direct consequence of $$ \ln(\ln(x))\leq \ln(x) $$ for large $x$.

Answer 1

Hint: What's the limit

$$\lim_{x\to \infty} \left(\frac{\frac{\ln (\ln(x))}{x}}{\frac{\ln(x)}{x}}\right)?$$

What does the value of this limit tell you about the quantity $$\frac{\frac{\ln (\ln(x))}{x}}{\frac{\ln(x)}{x}}?$$