By definition, asymptotic means that the process is a limiting one. So the point is reached "at t= \infty" althought this statement is not well defined mathematically. At no finite time will the state of the process actually be equal to the final point, although it can be said to be getting "closer" to the final point with time. Here the measure of closeness needs to be appropriately defined. There are several definitions of this measure and hence of stability.