`lim_(a to oo) (sin ""(pi)/(2n) sin ""(2pi)/(2n) sin""(3pi)/(2n).....sin ""((n-1)pi)/(n))^(1//n)` is equal to -

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. none of these

Answer

A

Explanation

`y=underset(ntooo)lim (sin""(pi)/(2n).sin ""(2pi)/(2n),sin ""(3pi)/(2n)...sin"" ((n-1)pi)/(n))^(1//n)`
consider log both side
` log y= underset(n to oo)lim (1)/(n)(log sin ""(pi)/(2n)+log sin ""(2pi)/(2 n)+log sin ""(3pi)/(2n)+...+log sin ""((n-1)pi)/(2n))`
`= underset( n to oo ) lim sum_(r=1) ^(r=n-1)(1)/(n)log sin ""(r pi)/(2n)`
`=int _(0)^(pi//2) log sin ""(pi)/(2) x .dx " " t =(pi x)/(2) `
`log y= (2) /(pi)((-pi)/(2)log 2)=- log 2`
`= log2 ^(-1)`
`y=(1)/(2)`