Example: Let $x, y, z$ be 3 non-zero real numbers. Find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$.
Introducing the parameter $\beta$, by the AM-GM inequality (here $\beta \in \mathbf{R}^{+}$, and $\beta^{2}<1$)
$$
x^{2}+\beta^{2} y^{2} \geqslant 2 \beta x y,\left(1-\beta^{2}\right) y^{2}+z^{2} \geqslant 2 \sqrt{1-\beta^{2}} y z
$$

Thus, $x^{2}+y^{2}+z^{2} \geqslant 2\left(\beta x y+\sqrt{1-\beta^{2}} y z\right)$. Let $\frac{\beta}{\sqrt{1-\beta^{2}}}=\frac{1}{2}$, solving gives $\beta=\frac{\sqrt{5}}{5}$. Therefore, $x^{2}+y^{2}+z^{2} \geqslant \frac{2}{5} \sqrt{5}(x y+2 y z)$, which means $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}} \leqslant \frac{\sqrt{5}}{2}$. When $x=k, y=\sqrt{5} k, z=2 k$ (here $k \neq 0$), the equality holds, and $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$ achieves $\frac{\sqrt{5}}{2}$, thus its maximum value is $\frac{\sqrt{5}}{2}$.