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Discuss the continuity of the following ...

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)*`

Text Solution

Verified by Experts

`f((pi)/(3))=3" "..."Given"...(1)`
`underset(x to pi//2)limf(x)=underset(xto pi//2)lim(1-sin x)/(((pi)/(2)-x)^(2))`
Put `pi/2-x=theta. ` Then ` x=pi/2-theta and as x to (pi)/(2)*theta to 0`
`therefore underset(xto pi//2)limf(x)=underset(thetato 0)lim (1-sin ((pi)/(2)-theta))/(theta^(2))`
`underset(thetato 0)lim(1-cos theta)/(theta^(2))`
`=underset(thetato 0 )lim (1-cos theta)/(theta^(2))xx(1+cos theta)/(1+cos theta)`
`=underset(thetato 0)lim (1-cos^(2)theta)/(theta^(2)(1+cos theta))`
`=underset(thetato 0)lim (sin ^(2)theta)/(theta^(2))xx(1)/(underset(thetato 0)lim (1+cos theta))`
` =(1)^(2)xx(1)/(1+1)=1/2" "...(2)`
From (1) and (2), `underset(xtopi//2)limf(x)ne f ((pi)/(2))`
`therefore f` is discontinous at `x=(pi)/(2).`
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