A computational study was conducted on expanding spherical pre-mixed flames to investigate the dynamics of the stretch-affected, one-dimensional, pulsating instability at elevated pressures and temperatures. Rich H-2/air mixtures were investigated using a time-dependent, spherically symmetric code with detailed chemistry, transport, and radiation submodels. The study shows that the region of instability in the pressure versus equivalence ratio domain for outwardly propagating flames is significantly narrowed when compared with that for planar flames due to the presence of the local ignition limit despite stretch-induced pulsation modifying the other boundary. Moreover, a region of one oscillation ending in extinction (even under adiabatic conditions which are not present in planar flames) has been discovered. However, similar to planar flames, radiative loss in outwardly propagating spherical flames reduces the region of instability due to transient extinction. Interestingly, this limit lies next to and just below the concentration limit for ignition. For the same ignition/extinction reasons, the region of instability is expanded at elevated temperatures due to increased flammability. The calculations show that the dynamics of spherically expanding flames is significantly different than that for planar flames. As a result, the near-limit phenomenon is extremely geometry dependent, thereby having an impact on practical considerations of flammability and stability.