Let \( f(x)=x^{2}+a x+3 \) and \( g(x)=x+b \), where \( F(x)=\lim _{n \rightarrow \infty} \frac{...

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Let \( f(x)=x^{2}+a x+3 \) and \( g(x)=x+b \), where \( F(x)=\lim _{n \rightarrow \infty} \frac{f(x)+\left(x^{2}\right)^{n} g(x)}{1+\left(x^{2}\right)^{n}} \). If \( F(x) \) is continuous at \( x=1 \) and \( x=-1 \) then find the value of \( \left(a^{2}+b^{2}\right) \).
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