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Question

The function F(x), defined as F(x)=limnf(x)+x2ng(x)1+x2n shall be continuous everywhere, if

A
f(1)=g(1)
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B
f(1)=g(1)
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C
f(1)=g(1)
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D
f(1)=g(1)
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Solution

The correct options are
A f(1)=g(1)
B f(1)=g(1)
We have,
F(x)=f(x),|x|<1 [limnx2n=0 for |x|<1]
f(x)+g(x)2,|x|=1
=limnx2nf(x)+g(x)x2n+1=g(x),|x|>1
[=limnx2n=0 for |x|>1]
Thus, the function F(x) shall be continuous everywhere if f(x) and g(x) are continuous everywhere and if F(x) is continuous at x=±1, we have
f(1)=f(1)+g(1)2=g(1)f(1)=g(1)
and, f(1)=f(1)+g(1)2=g(1)f(1)=g(1)

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