The value of `(lim)_(nvecoo)((sqrt(n^2+n)-1)/n)^2sqrt(n^(2+n)-1)` is `e` b. `1//e` c. `e^2` d. `e^(-2)`

Options

A. e
B. `1//e`
C. `e^(2)`
D. `e^(-2)`

Answer

B

Explanation

`L=underset(nrarroo)(lim)((sqrt(n^(2)+n)-1)/(n))^(2sqrt(n^(2)+n)-1)`
`=e^(underset(nrarroo)(lim)[(sqrt(n^(2)+n)-1-n)/(n)](2sqrt(n^(2)+n)-1))`
`=e^(underset(nrarroo)(lim)([(n^(2)+n)-(1+n)^(2)](2sqrt(n^(2)+n)-1))/(n{sqrt(n^(2)+n)+(1+n)}))`
`=e^(underset(nrarroo)(lim)((-n-1)(2sqrt(n^(2)+n)-1))/(n{sqrt(n^(2)+n)+(1+n)}))`
`=e^(underset(nrarroo)(lim)((-1-(1)/(n))(2sqrt(1+(1)/(n))-(1)/(n))n^(2))/(n^(2){sqrt(1+(1)/(n))+(1)/(n)+1})=e^(-1))`