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Evaluate lim(xto0) {1^(1//sin^(2)x)+2^(1...

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

Text Solution

Verified by Experts

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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