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  4. displaystyle lim t arrow 0(1(1/ sin 2 t)+2(1/ sin 2 t)+ ldots+n(1/ sin 2 t)) sin 2 t is equal to

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Q. $\displaystyle\lim _{t \rightarrow 0}\left(1^{\frac{1}{\sin ^2 t}}+2^{\frac{1}{\sin ^2 t}}+\ldots+n^{\frac{1}{\sin ^2 t}}\right)^{\sin ^2 t}$ is equal to

JEE MainJEE Main 2023Limits and Derivatives

Solution:

$ \displaystyle\lim _{t \rightarrow 0}\left(1^{\text{cosec}^2 t}+2^{\text{cosec}^2 t}+\ldots \ldots . .+n^{\text{cosec}^2 t}\right)^{\sin ^2 t} $
$ =\displaystyle\lim _{t \rightarrow 0} n\left(\left(\frac{1}{n}\right)^{\text{cosec}^2 t}+\left(\frac{2}{n}\right)^{\text{cosecs}^2 t}+\ldots \ldots . .+1\right)^{\sin ^2 t} $
$ = n $