Question #41249

Let $A(t) = \begin{bmatrix} 1 & 1 & 1 \\ [t-1] & [t] & [t+1] \\ 2 & 3 & -1 \end{bmatrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $C_{11}(t), C_{12}(t), C_{13}(t)$ be the cofactors of the elements of the first row of $A(t)$. A point $P(t)$ in 3D space is defined by the coordinates $(C_{11}(t), C_{12}(t), C_{13}(t))$. Consider another matrix $B(t) = \begin{bmatrix} C_{11}(t) & C_{12}(t) & C_{13}(t) \\ 2 & -1 & 3 \\ 1 & 4 & -2 \end{bmatrix}$. Let $M_{11}, M_{12}, M_{13}$ be the minors corresponding to the elements $C_{11}(t), C_{12}(t), C_{13}(t)$ respectively in $B(t)$. Define a modified determinant function $\Delta^*(t)$ based on the octant location of $P(t)$ as: $\Delta^*(t) = \operatorname{sgn}(C_{11}(t)) M_{11} + \operatorname{sgn}(C_{12}(t)) M_{12} + \operatorname{sgn}(C_{13}(t)) M_{13}$ where $\operatorname{sgn}(\cdot)$ is the signum function. Find the value of $\lim_{t \to 0^+} \Delta^*(t) - \lim_{t \to 0^-} \Delta^*(t)$.