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

Options

A. `1+sqrt5`
B. `-1+sqrt5`
C. `-1+sqrt2`
D. `1+sqrt2`

Answer

B

Explanation

Let `l = lim_(n to oo) 1/n overset(2n)underset(r=1) r/sqrt(n^(2)+r^(2))`
`lim_(n to oo)1/n overset(2n)underset(r=1)sum r/(nsqrt(1+(r//n)^(2)))`
` = lim_( n to oo) 1/n overset(2n) underset(r=1)sum(r//n)/(sqrt(1+(r//n)^(2)))`
Put ` r/n xx , 1/n= dx`
` lim_(n to oo) overset(2n) underset(r=1)sum = int_(0)^(2) x/(sqrt(1+x^(2)))dx `
`= [sqrt(1+x^(2))]_(0)^(2)= sqrt5=1`