Question

Consider $f\left(x\right)=x^2+ax+3$ and $g\left(x\right)=x+b$ and $F\left(x\right)=\underset{n\to \infty }{lim}\frac{f\left(x\right)+x^{2n}g\left(x\right)}{1+x^{2n}}$

If $F\left(x\right)$ is continuous at $x=1$, then

• A. $b=a+3$
• B. $b=a-1$
• C. $a=b-2$
• D. none of these

Answer 1

The correct option is A $b=a+3$

$F\left(x\right)=\frac{f\left(x\right)+x^{2n}g\left(x\right)}{1+x^{2n}}$

$F\left(x\right)=f\left(x\right)for0<x^2<1=\left(f\right(x)+g(x\left)\right)/2for{x}^2=1=g\left(x\right)for{x}^2>1$

$F\left(x\right)=g\left(x\right)forx<-1=\frac{g(-1)+f(-1)}{2}forx=-1=f\left(x\right)for-1<x<1=\frac{g\left(1\right)+f\left(1\right)}{2}forx=1=g\left(x\right)forx>1$

$forcontinuityatx=1$

$f\left(1\right)=g\left(1\right)$

$1+a+3=b+1$

$b-a=3$

$b=a+3$