An object of mass \(m\) is released from rest at time \(t = 0\) and falls vertically through a fluid that exerts a resistive force of magnitude \(F_d = bv\), where \(b\) is a positive constant and \(v\) is the speed of the object.

The downward speed of the object as a function of time \(t\) is given by
\[ v(t) = \dfrac{mg}{b}\left(1 - e^{-\frac{bt}{m}}\right) \]
where \(g\) is the acceleration due to gravity.

Which of the following correctly describes the limiting behavior of the speed \(v(t)\) as \(b \to 0\) at a fixed time \(t\), and the acceleration \(a(t)\) as \(t \to \infty\)?