A high-speed research probe moves along a straight line through a resisting medium that slows it down according to the velocity function \(v(t) = v_0 e^{-kt}\), where \(v_0\) is the speed at time \(t = 0\) and \(k\) is a positive constant. The probe passes position \(x = 0\) at \(t = 0\). Which of the following statements correctly describes the limiting distance traveled by the probe as \(t \to \infty\) and the time \(t_{0.9}\) required for the probe to cover \(90\%\) of that total limiting distance?