A flywheel with rotational inertia \(I\) is mounted on a fixed frictionless axle and rotates freely with an initial angular speed \(\omega_0\). At time \(t = 0\), a magnetic brake is engaged, exerting a retarding torque of magnitude \(\tau(t) = \tau_0 e^{-t/T}\) that opposes the rotation, where \(\tau_0\) and \(T\) are positive constants satisfying \(\tau_0 T < I \omega_0\). Which of the following statements correctly describes the angular speed \(\omega(t)\) and total angular displacement \(\Delta\theta(t)\) of the flywheel in the limit as \(t \to \infty\)?