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The value of lim(n rarr oo)(1/sqrt(4n^(2...

The value of `lim_(n rarr oo)(1/sqrt(4n^(2)-1)+1/sqrt(4n^(2)-4)+...+1/sqrt(4n^(2)-n^(2)))` is -

A

`1/4`

B

`pi/12`

C

`pi/4`

D

`pi/6`

Text Solution

Verified by Experts

The correct Answer is:
D
`lim_(n rarr oo) sum_(r=1)^(n) (1)/sqrt(4n^(2)-r^(2))=lim_(n rarr oo)1/n sum_(r=1)^(n) (1)/sqrt(4-(r//n)^(2))`
`=underset(0)overset(1)(int) (dx)/sqrt(4-x^(2))=(sin^(-1) (x/2))_(0)^(1)=pi/6`
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