Limit Level 1 -
Braver (start with by default).
Cross-slash (use Braver 8 times).
Limit Level 2 -
Blade Beam (kill a total of 120 enemies, all kills count towards this starting from the very first battle).
Climhazzard (use Blade Beam 7 times).
Limit Level 3 -
Meteorain (kill 200 enemies, kills begin to count after unlocking Blade Beam).
Finishing Touch (use Meteorain 6 times).
Limit Level 4 -
Omnislash (learn from Omnislash manual earned from the Gold Saucer's Battle Square).
Omnislash manual can be obtained when the Tiny Bronco becomes available for 51,200 BP if you prefer grinding it out very early. Alternatively, after acquiring the Highwind, it will only cost 32,000 BP and by then you'll be much stronger and breeze through the battles.