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Question

If f(x) =sin3x+Asin2x+Bsinxx5(x0) is continuous at x=0, then find A +B.

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Solution

f(x)=limx0sin3x+Asin2x+Bsinxx5

=limx0sinxx(34sin2x+2Acosx+Bx4)

=limx01+2cos2x+2Acosx+Bx4

=limx01x4(1+2(1(2x)22!+(2x)44!+.....)+2A(1x22!+x44!)+B)

=limx01x4(3+2A+B+x2(4A)+x4(43+A12)+.....)

2A+B+3=0 and 4A=0
A=4,B=5

A+B=4+5=1

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