લક્ષ અને વિકલન — ગણિત ધોરણ 11 સાયન્સ — Question

$=\lim_{n \rightarrow \infty}\ n^{\frac{2}{3}}\left[\left(1+\frac{1}{n}\right)^\frac{2}{3}-\left(1-\frac{1}{n}\right)^{\frac{1}{3}}\right] \ \ \ \ \ ......... (1)$

$\left(1+\frac{1}{n}\right)^{\frac{2}{3}}=1+\frac{2}{3}×\frac{1}{n}+\frac{\frac{2}{3}\left(\frac{2}{3}-1\right)}{2!}×\frac{1}{n^2}+....$

$\left(1-\frac{1}{n}\right)^{\frac{1}{3}}=1-\frac{2}{3}×\frac{1}{n}+\frac{\frac{2}{3}\left(\frac{2}{3}-1\right)}{2!}×\frac{1}{n^2}-....$

$\therefore \left(1+\frac{1}{n}\right)^{\frac{2}{3}}-\left(1-\frac{1}{n}\right)^{\frac{2}{3}}=\frac{4}{3}×\frac{1}{n}+\frac{8}{81}×\frac{1}{n^3}+....$

$\therefore$ સમીકરણ $(1)$ પરથી

$\lim_{n \rightarrow \infty} n^{\frac{2}{3}}\left[\frac{4}{3}×\frac{1}{n}+\frac{8}{81}×\frac{1}{n^3}+......\right]$

$\lim_{n \rightarrow \infty} \left[\frac{4}{3}×\frac{1}{n^{\frac{1}{3}}}+\frac{8}{81}×\frac{1}{n^{\frac{7}{3}}}+......\right]$

$=0$