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Question

If limx0x(1+acosx)bsinxx3=1, then

A
ba=56
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B
a=52
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C
ab=1
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D
b=32
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Solution

The correct options are
B a=52
C ab=1
We have
limx0x(1+acosx)bsinxx3=1
limx0x(1+a(1x22!+x44!...))b(xx33!+x55!...)x3
limx0x(1+ab)+x3[a2!+b3!]+0(x4)x3
This limit exist if 1+ab=0 and is equal to 1 and if a/2+b/6=1. Solving these two equations, we get a=5/2 and b=3/2

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