Grand canonical transition-matrix Monte Carlo is used to investigate kinetic and thermodynamic stability limits of the liquid and vapor in the Lennard-Jones fluid. Kinetic spinodals are associated with a vanishing of the free-energy barrier between metastable and stable states, and thermodynamic spinodals are related to diverging thermodynamic response functions. We show that the kinetically and thermodynamically defined spinodals are conceptually equivalent viewpoints in the case of the liquid-vapor transition. In agreement with the work of others, we find that the properties of the supersaturated vapor at its limit of stability approach their saturation values with increasing system size. We also find an analogous trend in the case of the superheated liquid. Finally, the computational algorithm and manipulation of data outlined in this work provide an intuitive and efficient method for determining thermophysical properties of stable and metastable fluids via molecular simulation.