Q: The value of $\large lim_{x \rightarrow 0} [1^{1/sin^2 x} + 2^{1/sin^2 x} + …. + n^{1/sin^2 x}]^{sin^2 x} $
(A) ∞
(B) 0
(C) $\frac{n(n+1)}{2}$
(D) n
Solution : Put $\frac{1}{sin^2 x} = t \ge 1 $
$\large lim_{t \rightarrow \infty} [1^t + 2^t + …. + n^t]^{1/t} $
$\large lim_{t \rightarrow \infty} n[(\frac{1}{n})^t + (\frac{2}{n})^t + …. + 1]^{1/t} $
= n[0 + 0 + … + 1]0 = n
Hence (D) is the correct answer.