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Evaluate: ("lim")(n rarr oo)(1/(sqrt(4n^...

Evaluate: `("lim")_(n rarr oo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))+ ........+1/(sqrt(3n^2)))`

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Verified by Experts

The correct Answer is:
`(pi)/6`
Given limit
`=lim_(nto oo) 1/n[n/(sqrt(4n^(2)-1))+n/(sqrt(4n^(2)-2^(2)))+…………+n/(sqrt(4n^(2)-n^(2)))]`
`=lim_(n to oo) 1/n[1/(sqrt(4-(1/n)^(2)))+1/(sqrt(4-(2/n)^(2)))+……………1/(sqrt(4-(n/n)^(2)))]`
`=int_(0)^(1)(dx)/(sqrt(4-x^(2)))=|sin^(-1)x/2|_(0)^(1)="sin"^(-1)1/2-0=(pi)/6`
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