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Question

Let f(x)=x2+ax+3,g(x)=x+b and F(x)=limnf(x)+x2ng(x)1+x2n. If F(x) is continuous xR, 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=4
F(x)=limnf(x)+x2ng(x)1+x2nF(x)=limnf(x)+(x2)ng(x)1+(x2)nF(x)=⎪ ⎪ ⎪⎪ ⎪ ⎪f(x), 0x2<1f(x)+g(x)2, x2=1g(x), x2>1F(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 xR, then
F(x) must be continuous at x=±1

For continuity at x=1
f(1)=g(1)=f(1)+g(1)24a=b1a+b=5(1)

For continuity at x=1
f(1)=g(1)=f(1)+g(1)21+a+3=1+bab=3(2)

Using equations (1) and (2), we get
a=1, b=4

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