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Question

Find the value of p and q for which
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1sin3x3cos2x ;ifx<x2p ;ifx=π2q(1sinx)(π2x)2;ifx>π2 is continuous at x=π2.

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Solution

left hand limit at x=π/2,limxπ/2f(x)=limx0f(π2h)
=limh01sin3(π2h)3cos2(π2h)=1cos3(h)3sin2h=00 form
By l hospital, limh03cos2h(sinh)3×2sinhcosh=12
For f(n) to be continuous, p=12
Right hand limit at x=π/2,limxπ/2+f(x)=limh0f(π2+h)
=limh0q(1sin(π2+h))(π2(π2h))2=q(1cosh)4h2
As cosh=12sin2h/2limh0q(2sin2h/2)h2/4.4
=limh0q4×2(sinh/2h/2)2=q2
q2=12q=1

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