The "Limited" restriction refers to playing cards from your hand. Once they are in the play area, there is no restriction on their usage.
"Play" in this game (and in most, generally) refers to paying the resource cost, and putting the card from your hand into the play area (I'm ignoring the difference for Event cards, which are played, resolved and then discarded). And, in general, once a card has been played you can use it immediately. So, in your example, you play one Sith Library on your first turn. You may use it to generate a resource during that turn, if you desire. But you may not play another "Limited" card during the turn. Next turn, you can play another Sith Library, and use it immediately. So, technically you can use both Sith Libraries on the second turn, and do not have to wait until the third.
In a game situation recently I had three Sith Libraries in my opening hand. I would have liked to pay cost, play one, use the first library to generate a resource to play the second, use the second to generate a resource to play the third! BOOM! for a net cost of one resource I would have three more resources next turn! Alas, Sith Library is Limited to one per turn in order to prevent just such a game changing power grab. You must build up over time.