Q.

Let f(x)=-11(1-|t|)cos(xt)dt then which of the following hold true?

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a

limx0f(x) exits and is equal to 1

b

fx is continuous at x = 0

c

f(0) is not defined. 

d

limx0f(x) exits and is equal to 2 

answer is C, D.

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Detailed Solution

f(x)=201(1-t)Icos(xt)IIdt=[(1t)sinxtx|10+1x01sinxtdt]=2[01x2cosxt|01]

f(x)=2[1cosxx2]                            (x0)

f(0)=-11(1-|t|)dt=201(1-t)dt=1

Hence f(x)=[2(1cosxx2)if  x01if  x=0

 f is continuous at x = 0 .

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