Discuss the continuity of the following functions at the points shown against them :
`{:(f(x)=(1-sinx)/(((pi)/(2)-x)^(2))",", "for"x ne(pi)/(2)),(=3",","for"x=(pi)/(2)):}}at x=(pi)/(2)*`

Explanation

R.H.L. `=lim_(hto0)f((pi)/(2)+h)=lim_(hto0)(1-sin((pi)/(2)+h))/(((pi)/(2)-(pi)/(2)-h)^(2))`
`=lim_(hto0)(1-cosh)/((-h)^(2))`
`=lim_(hto0)(1-1+2sin^(2)((h)/(2)))/((-h)^(2))`
`=lim_(hto0)(2sin^(2)((h)/(2)))/(h^(2))`
`=lim_(hto0)(2sin^(2)((h)/(2)))/(4((h)/(2))^(2))`
`=(1)/(2)lim_(hto0)(("sin"(h)/(2))/((h)/(2)))^(2)`
`=(1)/(2)xx1=(1)/(2)`
L.H.L. `=lim_(hto0)f((pi)/(2)-h)`
`lim_(hto0)(1-sin((pi)/(2)-h))/(((pi)/(2)-(pi)/(2)+h)^(2))`
`=lim_(hto0)(1-cosh)/(h^(2))`
`=(1)/(2)`
`=(1)/(2)ne3`
R.H.L=L.H.L `ne((pi)/(2))`
So,function is discontinuous at `x=(pi)/(2)`.