A particle moves along the x-axis starting from rest at the origin, with \(x(0) = 0\) and \(v(0) = 0\) at time \(t = 0\). For \(t \ge 0\), its acceleration is modeled by \(a(t) = a_0 e^{-\beta t}\), where \(a_0\) and \(\beta\) are positive constants. Which of the following statements correctly characterizes the limiting behavior of the particle's position \(x(t)\)?