The value of `lim_(ntooo) [root(3)((n+1)^(2))-root(3)((n-1)^(2))]` is __________.

Options

A. Equals `(1)/(sqrt(2))`
B. Does not exist
C. Equals `sqrt(2)`
D. Equals -sqrt(2)`

Answer

`(0)`

Explanation

`underset(ntooo)lim[root(3)((n+1)^(2))-root(3)((n-1)^(2))]`
`=underset(ntooo)limn^(2//3)[(1+(1)/(n))^(2//3)-(1-(1)/(n))^(2//3)]`
`=underset(ntooo)limn^(2//3)[(1+(2)/(3).(1)/(n)+((2)/(3)((2)/(3)-1))/(2!)(1)/(n^(2))...)-(1-(2)/(3).(1)/(n)+((2)/(3)((2)/(3)-1))/(2!)(1)/(n^(2))...)]`
`=underset(ntooo)limn^(2//3)[(4)/(3).(1)/(n)+(8)/(81).(1)/(n^(3))+...]`
`=underset(ntooo)lim[(4)/(3).(1)/(n^(1//3))+(8)/(81).(1)/(n^(7//3))+...]=0`