Ltt→0(11sin2t+21sin2t+ .......+n1sin2t)sin2t=

lLtt→0(11sin2t+21sin2t+ ......+n1sin2t)sin2t =ltt→0(1cosec2t+2cosec2t+......+ncosec2t)sin2t =ltt→0(ncosec2t)sin2t[(1n)cosec2t+(2n)cosec2t+...+1]sin2t =n[0+0+.....+1]0=n

Ltt→0(11sin2t+21sin2t+ .......+n1sin2t)sin2t=

lLtt→0(11sin2t+21sin2t+ ......+n1sin2t)sin2t =ltt→0(1cosec2t+2cosec2t+......+ncosec2t)sin2t =ltt→0(ncosec2t)sin2t[(1n)cosec2t+(2n)cosec2t+...+1]sin2t =n[0+0+.....+1]0=n

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

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$n^{2} + n$

$\frac{n (n + 1)}{2}$

$n^{2}$

answer is B.

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Detailed Solution

l $\underset{t \to 0}{L t} {(1^{\frac{1}{\sin^{2} t}} + 2^{\frac{1}{\sin^{2} t}} + ...... + n^{\frac{1}{\sin^{2} t}})}^{\sin^{2} t} = l t_{t \to 0} {(1^{\cos e c^{2}} t + 2^{\cos e c^{2}} t + . . . . . . + n^{\cos e c^{2}} t)}^{\sin^{2} t} = l t_{t \to 0} (n^{\cos e c^{2}} t)^{\sin^{2} t} [{(\frac{1}{n})}^{\cos e c^{2} t} + {(\frac{2}{n})}^{\cos e c^{2} t} + . . . + 1]^{\sin^{2} t} = n {[0 + 0 + . . . . . + 1]}^{0} = n$