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Q.

Which of the following functions are continuous on (0,π)

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a

xsinx,0<xπ2π2sin(π+x),π2<x<π

b

0xtsin1tdt

c

tanx

d

1,0<x3π42sin29x,3π4<x<π

answer is B, C.

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Detailed Solution

On(0,π)

 a) f(x)=tanxWe know that tan x is discontinuous  at x=π/2

 b) f(x)=0xtsin1tdt or f(x)=xsin1xf(x) differentiable, continuous on (0,π)

 c) L.H.L. =limx3π4(1)=1 R.H.L=limx3π42sin2x9=1

 Also, f3π4=1.f(x) is continuous on (0,π)

 d) L.H.L. =limxπ2xsinx=π2 R.H.L=limxπ2 π2sin(π+x)=-π2

f(x) is  not continuous

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