`("lim")_(xvec1)((1-x)(1-x^2)(1-x^(2n)))/({(1-x)(1-x^2)(1-x^n)}^2),n in N ,e q u a l s` `^2n P_n` (b) `^2n C_n` (c) `(2n)!` (d) none of these

Options

A. 0
B. `pi`
C. `2pi`
D. none of these

Answer

B

Explanation

`underset(xto1)lim((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2))`
`=underset(xto1)lim(((1-x)/(1-x))((1-x^(2))/(1-x))...((1-x^(2n))/(1-x)))/((((1-x)/(1-x))((1-x^(2))/(1-x))...((1-x^(n))/(1-x)))^(2))`
`=(1xx2xx3...(2n))/((1xx2xx3...n)^(2))=((2n)!)/(n!n!)=^(2n)C_(n)`