Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 28

Determine the $\displaystyle \lim_{x \to \infty} \frac{(\ln x)^2}{x}$. 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 x)^2}{x} &= \lim_{x \to \infty} \frac{2(\ln x)\left( \frac{1}{x} \right)}{1}\\
\\
&= \lim_{x \to \infty} \frac{2\ln x}{x}

\end{aligned}
\end{equation}
$


If we evaluate the limit, we will still get an indeterminate form. So, by applying L'Hospital's Rule once more...


$
\begin{equation}
\begin{aligned}
\lim_{x \to \infty} \frac{2 \ln x}{x} &= \lim_{x \to \infty} \frac{2 \left( \frac{1}{x} \right)}{1}\\
\\
&= \lim_{x \to \infty} \frac{2}{x}\\
\\
&= \frac{2}{\infty}\\
\\
&= 0
\end{aligned}
\end{equation}
$