Question #41211

Let $A(x)$ and $B(x)$ be $3 \times 3$ matrices defined for $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ as follows: $A(x) = \begin{bmatrix} x + x^2 \sin\left(\frac{1}{x}\right) & \sin x & \cos x \\ 0 & \cos x & \sin x \\ 0 & -\sin x & \cos x \end{bmatrix} \text{ for } x \neq 0, \text{ and } A(0) = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ $B(x) = \begin{bmatrix} \sec x & \tan x & 0 \\ \tan x & \sec x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ If $M(x) = (A(x) B(x))^T$, then the value of $\lim_{x \to 0} \frac{\det(M(x))}{e^{2x} - 1}$ is: