If `f(x)={(2cos x ,s in2x)/((pi-2x)^2),xlt=pi/2(e^(-cotx)-1)/(8x-4pi),x >pi/2` , then which of the following holds? `f` is continuous at `x=pi//2` `f` has an irremovable discontinuity at `x=pi//2` `f` has a removable discontinuity at `x=pi//2` None of these

Options

A. f is continuous at `x = pi//2`
B. f has an irremovable discontinuity at `x=pi//2`
C. f has a removable discontinuity at `x=pi//2`
D. none of these

Answer

B

Explanation

`f(x)={{:((2cosx-sin2x)/((pi-2x)^(2))",",x le(pi)/(2)),((e^(-cos)-1)/(8x-4pi)",",x gt(pi)/(2)):}`
L.H.L. at `x=pi//2`
`underset(hrarr0)(lim)(2sinh-sin2h)/(4h^(2))=underset(hrarr0)(lim)(2sinh(1-cosh))/(4h^(2))=0`
`"R.H.L."=underset(hrarr0)(lim)(e^(sinh)-1)/(8((pi//2)+h)-4pi)`
`=underset(hrarr0)(lim)(e^(sin h-1))/(8h).(sin h)/(sin h)=(1)/(8)`
`rArr" h(x) has irremovable discontinuity at x "=pi//2`.