8. There is a moving point $P$ on the $x$-axis, and fixed points $A(0,2), B(0,4)$. Then, when $P$ moves along the entire $x$-axis, the maximum value of $\sin \angle A P B$ is $\qquad$.
8. $\frac{1}{3}$
By symmetry, only consider the case where $P$ is on the positive x-axis:
Construct $\odot C$ passing through $A$ and $B$ and tangent to the positive x-axis, then the tangent point is the $P$ point that maximizes $\sin \angle A P B$. This is easily known from the fact that the inscribed angle on the same side of a fixed chord is greater than the angle outside the circle. (This is known from the fact that $\sin \angle A P B$ increases as $\angle A P B$ increases)
Connect $C P$, draw $C K \perp A B$ at $K$, then:
$$
\begin{array}{c}
O K=C P=3, \\
C K=\sqrt{A C^{2}-A K^{2}}=2 \sqrt{2}=O P, \\
\sin \angle O M P=\frac{O P}{B P}=\frac{2 \sqrt{2}}{2 \sqrt{6}}=\frac{\sqrt{3}}{3} . \\
\sin \angle A P B=\frac{A B}{A P}, \sin \angle O B P=\frac{1}{3} .
\end{array}
$$