Evaluate `lim_(ntooo) n^(2){sqrt((1-"cos"(1)/(n))sqrt((1-"cos"(1)/(n))sqrt((1-"cos"(1)/(n))...)))}`.

Explanation

Putting `(1)/(n)=x`,we have
`underset(xto0)lim(sqrt((1-cosx)sqrt((1-cosx)sqrt((1-cosx)...))))/(x^(2))`
`=underset(xto0)lim((1-cosx)^((1)/(2)+(1)/(2^(2))+(1)/(2^(3))+......oo))/(x^(2))`
`=underset(xto0)lim((1-cosx)^((1//2)/(1-1//2)))/(x^(2))`
`=underset(xto0)lim((1-cosx))/(x^(2)).((1+cosx))/((1+cosx))`
`=underset(xto0)lim((sinx)/(x))^(2).underset(xto0)lim(1)/((1+cosx))`
`=(1)^(2).(1)/(1+1)=(1)/(2)`