Question

# Let f(x)=x2+ax+3,g(x)=x+b and F(x)=limn→∞f(x)+x2ng(x)1+x2n. If F(x) is continuous ∀x∈R, then

A
a=5,b=4
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B
a=1,b=3
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C
a=4,b=1
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D
a=1,b=4
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Solution

## The correct option is D a=1,b=4F(x)=limn→∞f(x)+x2ng(x)1+x2n⇒F(x)=limn→∞f(x)+(x2)ng(x)1+(x2)n⇒F(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩f(x), 0≤x2<1f(x)+g(x)2, x2=1g(x), x2>1⇒F(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩g(x), x<−1f(−1)+g(−1)2, x=−1f(x), −1<x<1f(1)+g(1)2, x=1g(x), x>1 If F(x) is continuous ∀x∈R, then F(x) must be continuous at x=±1 For continuity at x=−1 f(−1)=g(−1)=f(−1)+g(−1)2⇒4−a=b−1⇒a+b=5⋯(1) For continuity at x=1 f(1)=g(1)=f(1)+g(1)2⇒1+a+3=1+b⇒a−b=−3⋯(2) Using equations (1) and (2), we get a=1, b=4

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