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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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CENGAGE ENGLISH-DEFINITE INTEGRATION -CAE_TYPE
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  4. Lim(n->oo)[1/n^2 * sec^2 (1/n^2)+2/n^2 * sec^2 (4/n^2)+..............+...

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  6. Evaluate the following limit: lim(nto oo)(sum(r=1)^(n) sqrt(r)sum(r=...

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  7. Evaluate the following limit: lim(nto oo)[(n!)/(n^(n))]^(1//n)

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  8. P rov et h a t4lt=int1^3sqrt(3+x^2)lt=4sqrt(3)

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  9. If I1=int0^1 2^x^2,I2=int0^1 2^x^3dx ,I3=int1^2^x^2dx ,I4=int1^2 2^x^3...

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  10. IfII=int0^(pi//2)cos(sinx)dx ,I2=int0^(pi/2)sin(cosx)d ,a n dI3=int0^(...

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  11. Prove that pi/6<int0^1(dx)/(sqrt(4-x^2-x^3))<pi/(4sqrt(2))

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  12. Evaluate int(0)^(pi//2)|sinx-cosx|dx.

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  13. Evaluate: int(-1)^4f(x)dx=4a n dint2^4(3-f(x))dx=7, then find the val...

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  14. Evaluate int(1)^(5)sqrt(x-2)sqrt(x-1)dx.

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  15. Evaluate: int(-1)^3(tan^(-1)(x/(x^2+1))+tan^(-1)((x^2+1)/x))dx

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  16. Evaluate int(1)^(a)x.a^(-[log(e)x])dx,(agt1).Here [.] represents the g...

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  17. Evaluate int(1)^(e^(6))[(logx)/3]dx, where [.] denotes the greatest in...

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  18. Find the value of int(-1)^1[x^2+{x}]dx ,w h e r e[dot]a n d{dot} denot...

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  19. Evaluate:- int0^(pi)[cot x]dx ,w h e r e[dot] denotes the greatest in...

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  20. Prove that int0^x[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes ...

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