\(A=\dfrac{x^2+x+1}{x^2-x+1}\Rightarrow Ax^2-Ax+A=x^2+x+1\Rightarrow\left(A-1\right)x^2-\left(A+1\right)x+A-1=0\)
\(\Delta=\left(A+1\right)^2-4\left(A-1\right)\left(A-1\right)\ge0\)
\(\Leftrightarrow A^2+2A+1-4\left(A^2-2A+1\right)\ge0\Leftrightarrow-3A^2+10A-3\ge0\)
\(\Rightarrow\dfrac{1}{3}\le A\le3\)
\(\Rightarrow\left\{{}\begin{matrix}A_{min}=\dfrac{1}{3}\\A_{max}=3\end{matrix}\right.\)