`lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)),`n in N`, `equals`

Options

A. 0
B. `2n^(C)n`
C. `2n!`
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)`