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

Evaluate `lim_(xto0) {1^(1//sin^(2)x)+2^(1//sin^(2)x)+...+n^(1//sin^(2

1.	Evaluate `lim_(xto0) {1^(1//sin^(2)x)+2^(1//sin^(2)x)+...+n^(1//sin^(2)x)}^(sin^(2)x)`.
Answer» Putting `(1)/(sin^(2)x)=t(ge1)`,we have `underset(t tooo)lim(1^(t)+2^(t)+...+n^(t))^(1//t)` `=underset(t tooo)lim(n^(t))^(1//t)[((1)/(n))^(t)+((2)/(n))^(t)+...+1]^(1//t)` `=n underset(t tooo)lim[((1)/(n))^(t)+((2)/(n))^(t)+...+1]^(1//t)` `=n[0+0+...+1]^(0)` `=n`

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