Grammatically and traditionally, "Everyone caught his limit" is undoubtedly the correct version. Pragmatically, however, "Everyone caught their limit" is coming more and more to be acceptable, fueled by (1) common spoken usage and (2) concerns with 'sexist' language. Good luck in defending your position.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.