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L=underset(nrarroo)(lim)((sqrt(n^(2)+n)-...

`L=underset(nrarroo)(lim)((sqrt(n^(2)+n)-1)/(n))^(2sqrt(n^(2)+n)-1)`

A

e

B

`1//e`

C

`e^(2)`

D

`e^(-2)`

Text Solution

Verified by Experts

The correct Answer is:
B
`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))`
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