Question

Evaluate : $\underset{x\to \infty }{lim}\frac{\ln x}{x^3}=$

Answer 1

Given,

${lim}_{x\to \infty }\left(\frac{\ln \left(x\right)}{x^3}\right)$

apply L-Hospital's rule,

$={lim}_{x\to \infty }\left(\frac{\frac{1}{x}}{3x^2}\right)$

$={lim}_{x\to \infty }\left(\frac{1}{3x^3}\right)$

$=\frac{1}{3}\times {lim}_{x\to \infty }\left(\frac{1}{x^3}\right)$

As $x\to \infty \Rightarrow \frac{1}{x}\to 0$

$=\frac{1}{3}\times {lim}_{\frac{1}{x}\to \infty }\left(\frac{1}{x^3}\right)$

$=\frac{1}{3}\times 0$

$=0$