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Evaluate lim(ntooo) (1)/(n^(2(log(e)n-lo...

Evaluate `lim_(ntooo) (1)/(n^(2(log_(e)n-log_(e)(n+1)))+n)`.

Text Solution

Verified by Experts

`underset(ntooo)lim(1)/(n^(2(log_(e)n-log_(e)(n-1)))+n)`
`=underset(ntooo)lim(1)/(n^(2)("log"_(e)(n)/(n+1))+n)`
`=underset(ntooo)lim(1)/(n^(2)("log"_(e)(1)/(1+(1)/(n)))+n)`
`=underset(xto0)lim(1)/((1)/(x^(2))("log"_(e)(1)/(1+x))+(1)/(x))`
`=underset(xto0)lim(1)/(-(log_(e)(1+x))/(x^(2))+(1)/(x))`
`=underset(xto0)lim(x^(2))/(x-log_(e)(1+x))`
`=underset(xto0)lim(x^(2))/(x-(x-(x^(2))/(2)))=2`
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