Question

Please Evaluate the limit​

Answer 1

Step-by-step explanation:

$\mathop{\lim}\limits_{{n}\mathrm{\rightarrow}\mathrm{\infty}}$ $\frac{{1}^{2}\mathrm{{+}}{2}^{2}\mathrm{{+}}{3}^{2}{\mathrm{....}}{n}^{2}}{{n}^{2}}$

$\mathop{\lim}\limits_{{n}\mathrm{\rightarrow}\mathrm{\infty}}$ $\frac{\sum{{n}^{2}}}{{n}^{3}}$

$\mathop{\lim}\limits_{{n}\mathrm{\rightarrow}\mathrm{\infty}}$ $\frac{{n}{\mathrm{(}}{n}\mathrm{{+}}{1}{\mathrm{)(}}{2}{n}\mathrm{{+}}{1}{\mathrm{)}}}{6{n}^{3}}$

$\mathop{\lim}\limits_{{n}\mathrm{\rightarrow}\mathrm{\infty}}$ $\frac{{2}{n}^{3}\mathrm{{+}}{3}{n}^{2}\mathrm{{+}}{n}}{6{n}^{3}}$

= $\frac{2}{6}$ = $\frac{1}{3}$

$Ans.$