Question

1. \lim _{x \

ightarrow \infty} \frac{\ln \ln x}{\sqrt{x}}

Explanation:

Step1: Apply L'Hopital's Rule

When \(x\to+\infty\), \(\lim_{x\to+\infty}\frac{\ln\ln x}{\sqrt{x}}\) is in the \(\frac{\infty}{\infty}\) form.
By L'Hopital's Rule, \(\lim_{x\to+\infty}\frac{\ln\ln x}{\sqrt{x}}=\lim_{x\to+\infty}\frac{\frac{1}{x\ln x}}{\frac{1}{2\sqrt{x}}}\)
Simplify the expression: \(\lim_{x\to+\infty}\frac{2}{ \sqrt{x}\ln x}\)

Step2: Analyze the limit

As \(x\to+\infty\), \(\sqrt{x}\to+\infty\) and \(\ln x\to+\infty\)
So, \(\lim_{x\to+\infty}\frac{2}{ \sqrt{x}\ln x}=0\)

Answer:

\(0\)