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Question

Let (x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩a|x2−x−2|2+x−x2,x<2b,x=2([.] denotes the greatest integer function)x−[x]x−2,x>2
If f(x) is continuous at x = 2, then

A
a = 1, b = 2
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B
a = 1, b = 1
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C
a = 0, b = 1
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D
a = 2, b = 1
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Solution

The correct option is B a = 1, b = 1
f(x)=a|x2x2|2+xx2,x<2x[x]x2,x=2a(x2x2)2+xx2,x<2x[x]x2,x>2f(x)=⎪ ⎪⎪ ⎪a,x<2b,x=2x[x]x2,x>2
Check at x = 2
LHL=limx2f(x)=limh0f(2h)=limh0a=aRHL=limx2+f(x)=limh0f(2h)=limh02+h[2+h]2+h2=limh02+h22+h2=1


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