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Question

# Let f:R→R be a function defined by f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1−cosaxx2if x<0bif x=0√x√16+√x−4if x>0 If f is continuous in R, then b−a can be

A
8
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B
6
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C
2
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D
4
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Solution

## The correct option is D 4f is continuous in R ⇒f is continuous at x=0 also. limx→0−f(x)=limx→0+f(x)=f(0) limx→0+f(x)=limx→0+√x√16+√x−4 =limx→0+√x16+√x−16⋅(√16+√x+4) =limx→0+(√16+√x+4)=4+4=8 ∴b=8 limx→0−f(x)=limx→0−1−cosaxx2=8 ⇒limx→0−2sin2a2xx2=8 ⇒limx→0−⎛⎜ ⎜⎝sina2xx⎞⎟ ⎟⎠2=4 ⇒limx→0−⎛⎜ ⎜⎝sina2xa2x⎞⎟ ⎟⎠2×(a2)2=4 ⇒1×(a2)2=4 ⇒a=±4 For a=4,b=8, we have b−a=4

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