\(A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\)
\(\Leftrightarrow A^2=\frac{x^2y^2}{z^2}+\frac{y^2z^2}{x^2}+\frac{z^2x^2}{y^2}+2\left(x^2+y^2+z^2\right)\)
\(\Leftrightarrow2A^2=\left(\frac{x^2y^2}{z^2}+\frac{y^2z^2}{x^2}\right)+\left(\frac{y^2z^2}{x^2}+\frac{z^2x^2}{y^2}\right)+\left(\frac{x^2y^2}{z^2}+\frac{z^2x^2}{y^2}\right)+12\)
\(\ge2\left(x^2+y^2+z^2\right)+12=6+12=18\)
\(\Rightarrow A\ge3\)