Example 9. If $x \neq 0$, then
$$
\frac{\sqrt{1+x^{2}+x^{4}}-\sqrt{1+x^{4}}}{x}
$$

The maximum value is $\qquad$. $(1992$, National Junior High School Competition)
Analyzing this type of problem generally involves rationalizing the numerator to achieve reduction. Noting that the maximum value is obtained when $x>0$, the original expression can be transformed into:
$$
\begin{array}{l}
\sqrt{\left(x-\frac{1}{x}-0\right)^{2}+(0-\sqrt{3})^{2}} \\
-\sqrt{\left(x-\frac{1}{x}-0\right)^{2}+(0-\sqrt{2})^{2}} .
\end{array}
$$

By analogy with the distance formula between two points, the geometric meaning of the above expression is: the difference in distances from a moving point $P\left(x-\frac{1}{x}, 0\right)$ to two fixed points $A(0, \sqrt{3})$, $B(0, \sqrt{2})$. (As shown in the figure below)
$\because|A B| \geqslant$
$|P A|-|P B|$,
and $|A B|$
$=\sqrt{3}-\sqrt{2}, \quad B C C$
Therefore, when point $P$ is

at the origin, i.e., $x=$ $\qquad$
$\frac{1}{x}, x=1$,
$|P A|-|P B|$ reaches
the maximum value $(\sqrt{3}-\sqrt{2})$