`If("lim")_(xvecoo)(n .3^n)/(n(x-2)^n+n .3^(n+1)-3^n)=1/3, t h e nt h e` range of `x` is (where `n in N)dot` (2,5) (b) (1,5) (c) `-5,5)` (d) `(-oo,oo)`

Options

A. 1
B. `(2//3)^(1//2)`
C. `(3//2)^(1//2)`
D. `e^(1//2)`

Answer

A

Explanation

`underset(ntooo)lim(n.3^(n))/(n(x-2)^(n)+n.3^(n+1)-3^(n))=1/3`
or `underset(ntooo)lim(1)/(((x-2)^(n))/(3^(n))+3-(1)/(n))=1/3" "("Dividing "N^(r)" and "D^(r)" by "nxx3^(n))`
`"For "underset(ntooo)lim" to be equal to "1//3,`
`underset(ntooo)lim((x-2)/(3))^(n)to0`
`:." "2lexlt5`