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Evaluate: lim(n->oo)n^2{sqrt((1-cos(1/n)...

Evaluate:` lim_(n->oo)n^2{sqrt((1-cos(1/n))sqrt((1-cos(1/n)) sqrt((1-cos(1/n)) ..oo)))}dot`

Text Solution

Verified by Experts

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)`
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