Question #41310

Let $M(x) = \begin{pmatrix} \sin x & \cos x & \cos(x-\frac{\pi}{6}) \\ \sin 2x & \cos 2x & \cos(2x-\frac{\pi}{6}) \\ \sin 3x & \cos 3x & \cos(3x-\frac{\pi}{6}) + \sqrt{3}\sin x \end{pmatrix}$ and $N(x) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \sin x \end{pmatrix}$.

Let $S = \{ x \in [0, 2\pi] : \det(M(x)) \neq 0 \}$. For $x \in S$, define $g(x) = \det(M(x)^{-1} N(x))$.

A function $h: [0, 2\pi] \to \mathbb{R}$ is defined as: $h(x) = \begin{cases} \lim_{n \to \infty} \frac{1}{1 + |g(x)|^n} & \text{if } x \in S \\ 0 & \text{if } x \notin S \end{cases}$

Find the number of points in $[0, 2\pi]$ where $h(x)$ is discontinuous.