Question

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

• A. $n ^2$
• B. $n ^2+ n$
• C. $n$
• D. $\frac{n(n+1)}{2}$

Answer 1

Answer: (C)

$n$

Answer 2

t→0lim​(1cosec2t+2cosec2t+……..+ncosec2t)sin2t
=t→0lim​n((n1​)cosec2t+(n2​)cosecs2t+……..+1)sin2t
=n