Example 9 In a regular $n$-sided pyramid, the range of the dihedral angle formed by two adjacent side faces is ( ).
(A) $\left(\frac{n-2}{n} \pi, \pi\right)$
(B) $\left(\frac{n-1}{n} \pi, \pi\right)$
(C) $\left(0, \frac{\pi}{2}\right)$
(D) $\left(\frac{n-2}{n} \pi, \frac{n-1}{n} \pi\right)$
(1994, National High School Mathematics Competition)
Analysis: It is easy to know that when the plane angle approaches infinitely close to the vertex of the base regular $n$-sided polygon, its value tends to $\frac{(n-2) \pi}{n}$. When $n \rightarrow +\infty$, the plane angle $\rightarrow \pi$. Therefore, the correct choice is (A).