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`lim_(a to oo) (sin ""(pi)/(2n) sin ""(2pi)/(2n) sin""(3pi)/(2n).....sin ""((n-1)pi)/(n))^(1//n)` is equal to -A. `(1)/(2)`B. `(1)/(3)`C. `(1)/(4)`D. none of these |
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Answer» Correct Answer - A `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)` |
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