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Q.

If  fx=x-ex+cos2xx2,  x≠0,  is  contitnuous  at x=0,  then where [x] and {r} denote the greatest integer and fractional part functions, respectively.

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a

f0=52

b

f0=-2

c

f0=0

d

f0f0=-1.5

answer is D.

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Detailed Solution

limx→0  x-ex+1-1-cos2xx2       =limx→0  x--ex+1x2-1-cos2xx2      =limx→0 x+1-1+x+x22x2-2sin2xx2        Using  expansion  of  ex       =-12-2         =-52Hence,  for  continuity ,  f0=-52.Now,   f0=-3;  f0=-52=12.Hencee,  f0f0=-32=-1.5
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If  fx=x-ex+cos2xx2,  x≠0,  is  contitnuous  at x=0,  then where [x] and {r} denote the greatest integer and fractional part functions, respectively.