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Evaluate: ("lim")(nvecoo)(1/(sqrt(4n^2-1...

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

A

0

B

1

C

`pi/3`

D

`pi/6`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `lim_(ntoinfty)((1)/(sqrt(4n^(2)-1))+(1)/(sqrt(4n^(2)-2^(2)))+...+(1)/(sqrt(3n^(2))))`
`=lim_(ntoinfty)(1)/(n)[(n)/(sqrt(4n^(2)-2^(2)))+...+(n)/(sqrt(4n^(2)-n^(2)))]`
`=lim_(ntoinfty)(1)/(n)[(1)/(sqrt(4-((1)/(n))^(2)))+(1)/(sqrt(4-((2)/(n))^(2)))+...+(1)/(sqrt(4-((2)/(n))^(2)))]`
`-lim_(ntoinfty)sum_(r=1)^(n)(1)/(sqrt(4-((r)/(n))^(2)))*(1)/(n)`
`=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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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  2. lim(n->oo) (1^p+2^p+3^p+...........+n^p)/n^(p+1)

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  3. Evaluate: ("lim")(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt...

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  4. If f(x) = tanx-tan ^(3) x + tan^(5) x - tan ^(7) x + ... infty for o...

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  8. The value of int (1)^(2) (dx)/((x+1) sqrt((x^2)-1)) is

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  9. If f(x)=|(sin x+sin x 2x+sin3x,sin2x,sin3x),(3+4sinx,3,4sinx),(1+sinx,...

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  10. int (-1)^(1) (e^(x^(3)) +e^(-x^(3))) (e^(x)-e^(-x)) dx is equal to

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  11. underset(n to oo)lim((1)/(n)+(1)/(n+1)+...+(1)/(3n)) is equal to

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  12. Let f (x) = x – [x], for every real number x, where [x] is the greates...

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  13. The value of int (0)^(1) (x^(4) +1)/(x^(2)+1)dx is

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  18. The value of int (1)^(e) 10^(log(e)x) dx is equal to

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  19. If int (2)^(e) (1/(logx)-1/(logx)^(2))dx = a + b/(log2) , then

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  20. int (0)^(oo) (dx)/((x+ sqrt(x^(2)+1))^(3)) is equal to

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