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 $x$.

score 4 · Accepted Answer · answered Sep 11 '18 at 13:19

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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 $$\frac{\frac{\ln (\ln(x))}{x}}{\frac{\ln(x)}{x}}?$$

answered Sep 11 '18 at 13:19

Zestylemonzi

4,103

1

The limit is $0$ (using L'Hospital's rule) and hence the statement is true. – Rhjg Sep 11 '18 at 13:25
Yeah, nice one! – Zestylemonzi Sep 11 '18 at 13:29

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

1 Answers1