A uniform solid cylinder of mass \(M\) and radius \(R\) is mounted on a frictionless horizontal axle through its central axis and is initially at rest. At time \(t = 0\), a time-dependent force \(F(t) = F_0 e^{-kt}\) is applied tangentially to the outer rim of the cylinder, where \(F_0\) and \(k\) are positive constants. Which of the following expressions represents the angular speed of the cylinder as \(t \to \infty\)?