limx→0 1cosec2⁡x+2cosec2⁡x+…+ncosec2⁡xsin2⁡x is

A
1
B
1/n
C
n
D
0

Solution:

Let, $y = {cosec}^{2} x$

Required $limit = lim_{y \to \infty} {(1^{y} + 2^{y} + \dots n^{y})}^{1 / y} [0 / \infty form]$

$\begin{matrix} = lim_{y \to \infty} {(n^{y})}^{1 / y} {[{(\frac{1}{n})}^{y} + {(\frac{2}{n})}^{y} + \dots + {(\frac{n - 1}{n})}^{y} + 1]}^{1 / y} \\ = lim_{y \to \infty} n {[{(\frac{1}{n})}^{y} + {(\frac{2}{n})}^{y} + \dots + {(\frac{n - 1}{n})}^{y} + 1]}^{1 / y} \\ = n \cdot 1^{0} = n \cdot 1 = n \end{matrix}$

$lim_{x \to 0} {(1^{{cosec}^{2} x} + 2^{{cosec}^{2} x} + \dots + n^{{cosec}^{2} x})}^{\sin^{2} x}$ is

Solution:

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