A flat circular disk of radius \(R\) and total mass \(M\) has a non-uniform surface mass density given by \(\sigma(r) = \sigma_0 \left(1 + \beta \dfrac{r^2}{R^2}\right)\), where \(\sigma_0\) and \(\beta\) are positive constants and \(r\) is the radial distance from the center. The rotational inertia of the disk about an axis perpendicular to the disk through its center is
\[ I = \left(\dfrac{3 + 2\beta}{6 + 3\beta}\right) M R^2 \]
Which of the following statements correctly describes and physically justifies the behavior of \(I\) in a limiting case?