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Question

The sum of an infinite geometric series whose first term is the limit of the function f(x)=tanxsinxsin3x as x0 and whose common ratio is the limit of the function g(x)=1x(cos1x)2 as x1 is

A
13
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B
14
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C
12
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D
23
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Solution

The correct option is D 23
a=limx0tanxsinxsin3x
=limx01cosx1sin2x
limx01cosxcosxsin2x
=limx0(1cosx)cosx(1+cosx)(1cosx)
=limx01cosx(1+cosx)
=11.(2)=12
r=limx11x(cos1x)2
=12x2cos1x(11x2)
=limx1=1x24xcos1x
=limx111(2x)4.2[1x212xcos1x+x(1)1x2
limx114x1x2cos1x2xx1x2
limx114.x(cos1x)1x22xx
=limx114.x.2x(cos1x)1x22x
r=14(1.2102.1)=14
a=12,r=14
s=a1r=1/211/4=2434=23

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