Mathematics - Limits Question with Solution | TestHub

$L=\underset{t\to 0}{\lim }{\left(1^{\frac{1}{{\sin }^{2}t}}+2^{\frac{1}{{\sin }^{2}t}}+3^{\frac{1}{{\sin }^{2}t}}......n^{\frac{1}{{\sin }^{2}t}}\right)}^{{\sin }^{2}t}$

$\underset{t\to 0}{=\lim }{\left(1^{{\mathrm{cosec}}^{2}t}+2^{{\mathrm{cosec}}^{2}t}+3^{{\mathrm{cosec}}^{2}t}......n^{{\mathrm{cosec}}^{2}t}\right)}^{{\sin }^{2}t}$

$\underset{t\to 0}{=\lim } n^{\frac{{\sin }^{2}t}{{\sin }^{2}t}}{\left({\left(\frac{1}{n}\right)}^{{\mathrm{cosec}}^{2}t}+{\left(\frac{2}{n}\right)}^{{\mathrm{cosec}}^{2}t}+{\left(\frac{3}{n}\right)}^{{\mathrm{cosec}}^{2}t}......1\right)}^{{\sin }^{2}t}$

$=n\underset{t\to 0}{\lim } {\left(0+0+0......1\right)}^0=n$