The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {\root 3 \of {{{\left( {n + 1} \right)}^2}} - \root 3 \of {{{\left( {n - 1} \right)}^2}} } \right]$$       is :

Solution :
$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } \left[ {\root 3 \of {{{\left( {n + 1} \right)}^2}} - \root 3 \of {{{\left( {n - 1} \right)}^2}} } \right] \cr & = \mathop {\lim }\limits_{n \to \infty } {n^{\frac{2}{3}}}\left[ {{{\left( {1 + \frac{1}{n}} \right)}^{\frac{2}{3}}} - {{\left( {1 - \frac{1}{n}} \right)}^{\frac{2}{3}}}} \right] \cr & = \mathop {\lim }\limits_{n \to \infty } {n^{\frac{2}{3}}}\left[ {\left( {1 + \frac{2}{3}.\frac{1}{n} + \frac{{\frac{2}{3}\left( {\frac{2}{2} - 1} \right)}}{{2!}}\frac{1}{{{n^2}}}.....} \right) - \left( {1 - \frac{2}{3}.\frac{1}{n} + \frac{{\frac{2}{3}\left( {\frac{2}{3} - 1} \right)}}{{2!}}\frac{1}{{{n^2}}}.....} \right)} \right] \cr & = \mathop {\lim }\limits_{n \to \infty } {n^{\frac{2}{3}}}\left[ {\frac{4}{3}.\frac{1}{n} + \frac{8}{{81}}.\frac{1}{{{n^3}}} + .....} \right] \cr & = \left[ {\frac{4}{3}.\frac{1}{{{n^{\frac{1}{3}}}}} + \frac{8}{{81}}.\frac{1}{{{n^{\frac{7}{3}}}}} + .....} \right] \cr & = 0 \cr} $$