False. The Church did not teach about the Limbo. The Latin Church, after the Schism, added this to the original faith. Shall we assume that we now know more than the Apostles or the Church Fathers who didn't teach it? Don't you know that the faith was delivered in toto and not piecemeal?
There were some theologians who speculated, but the entire Church did not buy into these and therefore they were not binding. Sorry to burst your bubble.
Fine, answer my question then. What is the fate of unbaptized infants and why? Are they born with grace or not? Is there some mode of salvation for them besides Baptism?