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Question

If the functions f(x), defined below is continuous at x=0, find the value of k:
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪1cos2x2x2,x<0k,x=0x|x|,x>0

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Solution

f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪1cos2x2x2,x<0Kx=0x|x|x>0⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Given, f(x) is continuous at x=0
limx0f(x)=f(0)=k
limx0+f(x)=limx0+=x|x|
=limx0+xx=1
limx0f(x)=limx01cos2x2x2
(L hospital Rule)=limx0(+sin2x)(/22)/42x
=limx0sin2x2x
=1
So, limx0f(x)=1K=1.

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