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Question

The value of f(0) so that the function f(x)=1cos(1cos x)x4 is continuous everywhere is 


A
18
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B
12
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C
14
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D
none of these
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Solution

The correct option is B 18
limx0f(x)=f(0) for continuity.
limx0f(x)=limx01cos (1cos x)x4
cos x=1x22!+x44!+.......limx0f(x)=limx01[1(1cos x)22!+(1cos x)44!....]x4(Replacing x by (1cos x))limx0f(x)=limx0(1cos x)22! x4(1cos x)44! x4+....Also, limx0(1cos xx2)=12      (cos x=1x22!+x44!+.....)limx0f(x)=limx012.((1cos x)x2)2+124limx0(x22!x44!.........)4x4+H.O.T.Slimx0f(x)=12×limx0(1cos xx2)2+0+0+.....limx0f(x)=12×(12)2=12×14=18

Mathematics

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