Question

The value of $\underset{n\to \infty }{lim}[\sqrt[3]{3{(n+1)}^2}-\sqrt[3]{3{(n-1)}^2}]$ is

Answer 1

$\underset{n\to \infty }{lim}[\sqrt[3]{3{(n+1)}^2}-\sqrt[3]{3{(n-1)}^2}]$

$=\underset{n\to \infty }{lim}{n}^{\frac{2}{3}}[{(1+\frac{1}{n})}^{\frac{2}{3}}-{(1-\frac{1}{n})}^{\frac{2}{3}}]$
Using binomial theorem expansions
$=\underset{n\to \infty }{lim}{n}^{\frac{2}{3}}[\frac{4}{3}.\frac{1}{n}+\frac{8}{81}.\frac{1}{n^3}+.....]$
$=\underset{n\to \infty }{lim}[\frac{4}{3}\cdot \frac{1}{n^{\frac{1}{3}}}+\frac{8}{81}\cdot \frac{1}{n^{\frac{7}{3}}}+....]=0$