We study quitting games and define the concept of absorption paths, which is an alternative definition to strategy profiles that accomodates both discrete time aspects and continuous time aspects, and is parameterized by the total probability of absorption in past play rather than by time. We then define the concept of sequentially 0perfect absorption paths, which are shown to be limits of ε-equilibrium strategy profiles as ε goes to 0. We finally identify a class of quitting games that possess sequentially 0-perfect absorption paths.