Evaluate `underset(xto0)lim{1^(1//sin^(2)x)+2^(1//sin^(2)x)+...+n^(1//sin^(2)x)}^(sin^(2)x)`.

Options

A. `oo`
B. `0`
C. `n`
D. `(n(n+1))/(2)`

Answer

C

Explanation

We, have
`lim_(xto0) {1^((1)/(sin^(2)x)+)2^((1)/(sin^(2)x))+3^((1)/(sin^(2)x))+.....+n^((1)/sin^(2)x)}^(sin^2x)`
`lim_(xto0)n{((1)/(n))^((1)/sin^(2)x)+((2)/(h))^((1)/sin^(2)x)+......+((n-1)/n)^((1)/sin^(2)x)+1}^((1)/sin^(2)x)`
`=n(0+0+......+0+1}=n`.