In patch 4.5, the controversial Blue Mage job was released and is the first of what is referred to as a Limited Job. Design very much for solo play in mind, the Blue Mage is probably best considered its own PvE content. If you decide to try your hand at the new job/content you may find the following guides useful for you.
To be true to the FF lore, the Blue Mage does not learn its skills through leveling or job quests. Instead Blue Mages must learn their actions through enemy encounters. There is no need to be hit by the enemy actions. It only needs to be used by the enemies for a CHANCE to learn the action. As such, two level 50 Blue Mages are likely to have different sets of spells learned.
A Blue Mage only content, players will go through a series of challenging encounters. The Blue Mage must equip themselves with the right set of actions to survive them.