The model ballooning mode equation of Connor, Hastie and Taylor for large-aspect-ratio circular tokamaks is analysed in the limit of large pressure gradient, and corresponding expressions for stability boundaries are derived. In particular, it is found that for fixed radial wavenumbers infinite sequences of unstable bands exist and that minimizing over the radial wavenumbers leads to asymptotic merging between the neighbouring bands, which significantly modifies the second stability boundary compared with that for zero radial wavenumber.