Let f(x)=[(1-sinpix)/(1+cos2pix), x<1/2...

Let `f(x)=[(1-sinpix)/(1+cos2pix), x<1/2 and p , x=1/2 and sqrt(2x-1)/(sqrt(4+sqrt(2x-1))-2)` .Determine the value of p, if possible, so that the function is continuous at `x = 1/2`.

`1//4`

none of these

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Verified by Experts

The correct Answer is:

`f((1)/(2))=p`
`" "underset(xrarr(1^(-))/(2))(lim)(1-sinpix)/(1+cos(2pix))`
`" "=underset(xrarr(1^(-))/(2))(lim)(-pi cos (pix))/(-2pi sin (2pix))" (Using L' Hospital's Rule)"`
`" "=underset(xrarr(1-)/(2))(lim)(-pisin(pix))/(4picos(2pix))" (Using L'Hospital's Rule)"`
`" "=(1)/(4)`
`" "underset(xrarr((1)/(2))^(+))(lim)((sqrt(2x-1))/(sqrt(4+sqrt(2x-1))-2))`
`" "underset(xrarr(1+)/(2))(lim)((sqrt(2x-1))/(4+sqrt(2x-1)-4))(sqrt(4+sqrt(2x-1))+2)`
`" =4`
Since LH.L. `ne` R.H.L., value of p does not exist.

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