Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 18

Determine the $\displaystyle \lim_{x \to \infty} \frac{\ln \ln x}{x}$. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.

By applying L'Hospital's Rule,

$\begin{equation} \begin{aligned} \lim_{x \to \infty} \frac{\ln \ln x}{x} &= \lim_{x \to \infty} \frac{\frac{\frac{d}{dx}\ln(x)}{\ln(x)}}{\frac{d}{dx}(x)}\\ \\ &= \lim_{x \to \infty} \frac{\frac{\left(\frac{1}{x}\right)}{\ln x}}{1}\\ \\ &= \lim_{x \to \infty} \frac{1}{x \ln x}\\ \\ &= \frac{1}{\infty (\ln (\infty))}\\ \\ &= \frac{1}{\infty}\\ \\ &= 0 \end{aligned} \end{equation}$