`lim_(n to oo)(1)/(n)sum_(r=1)^(2n)(r )/(sqrt(n^(2)+r^(2)))` बराबर है

Options

A. `1+sqrt(5)`
B. `-1+sqrt(5)`
C. `-1+sqrt(2)`
D. `1+sqrt(2)`

Answer

B

Explanation

` lim_(x-> infty)1 /n sum_(r=1)^(2n) r/(sqrt(n ^ 2 + r ^ 2))`

`lim_(x-> infty)1 /n sum_(r=1)^(2n)r/sqrt(n^2(1+ (r ^ 2)/(n ^ 2)))`

`lim_(x-> infty)1 /n sum _(r=1)^(2n)(r/n)/((1 + (r ^ 2)/(n ^ 2)))`

`1/ n ->infty`

इसलिए `1/n= dx`

` r = n Rightarrow1/ooRightarrowx=0`

` Rightarrow r = 2n Rightarrow(2n)/n Rightarrowx = 2`

`int _0 ^ 2 (xdx) /(sqrt (1+x^ 2)`

` 1+x^ 2 =t`

` 2xdx = dt`

`1/ 2 int dt/( sqrt t)`

`1 /2 int t^ (-1/2) dt`

`1/ 2 * 2 * sqrt(t) `

` sqrt(1 + x ^ 2)`

`int_0^ 2( xdx)/( sqrt (1+x^ 2)) `

सीमा का प्रयोग करने पर

` sqrt( 1+4 ) - sqrt( 1+0)`

`sqrt (5 -1) `

`-1+sqrt5`