Evaluate the following limits :
`lim_(xrarr0)sqrt((1/2(1-cos2x))/x`

Explanation

`underset(xto0)limsqrt((1)/(2)(1-cos2x))/(x)=underset(xto0)limsqrt((1)/(2).2sin^(2)x)/(x)=underset(xto0)lim(|sinx|)/(x)`
Now `underset(xto0^(+))lim(|sinx|)/(x)=underset(xto0^(+))lim(sinx)/(x)=1`
`underset(xto0^(-))lim(|sinx|)/(x)=underset(xto0^(-))lim(-sinx)/(x)=-1`
Thus `underset(xto0^(+))lim(|sinx|)/(x)neunderset(xto0^(-))lim(|sinx|)/(x).`
So, `underset(xto0)limsqrt(1/2(1-cos2x))/(x)` does not exist.