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lim(nto oo)+(1)/(sqrt(n^(2)+n))+(1)/(sqr...

`lim_(nto oo)+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+...(1)/(sqrt(n^(2)+(n-1)n))` is equal to

A

`2+2sqrt(2)`

B

`2sqrt(2)-2`

C

`2sqrt(2)`

D

2

Text Solution

Verified by Experts

The correct Answer is:
B
Given , `lim_(ntoinfty)[(1)/(n)+(1)/(sqrt(n^(2)+n))+..+(1)/(sqrt(n^(2)+(n-1)n))]`
`=lim_(n to infty)[(1)/(n)+(1)/(nsqrt(1+(1)/(n)))+..+(1)/(nsqrt(1+((n-1))/(n)))]`
`=(1)/(n)lim_(ntoinfty)[1+(1)/(sqrt(1)+(1)/(n))+..+(1)/(sqrt(1+((n-1))/(n)))]`
`-lim_(ntoinfty)sum_(r=0)^(n-1)((1)/(sqrt(1+(r)/(n))))(1)/(n)`
`=int_(0)^(1)(dx)/(sqrt(1+x))=[2sqrt(1+x)]_(0)^(1)=2sqrt(2)-2`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  5. Evaluate: ("lim")(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt...

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

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

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

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

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