Example 4 If $x \neq 0$, find the maximum value of
$$
\frac{\sqrt{1+x^{2}+x^{4}}-\sqrt{1+x^{4}}}{x}
$$

(1992, National Junior High School Mathematics League)
Solution: Original expression
$$
\begin{array}{l}
= \pm\left(\frac{\sqrt{1+x^{2}+x^{4}}}{|x|}-\frac{\sqrt{1+x^{4}}}{|x|}\right) \\
= \pm\left(\sqrt{\frac{1}{x^{2}}+1+x^{2}}-\sqrt{\frac{1}{x^{2}}+x^{2}}\right) \\
= \pm \frac{1}{\sqrt{\left(x-\frac{1}{x}\right)^{2}+3}+\sqrt{\left(x-\frac{1}{x}\right)^{2}}+2} .
\end{array}
$$

For the original expression to have a maximum value, only $\left(x-\frac{1}{x}\right)^{2}=0$, at this time, the maximum value sought is $\frac{1}{\sqrt{3}+\sqrt{2}}=\sqrt{3}-\sqrt{2}$.
5 Using $(a-b)^{2} \geqslant 0$