Limbo, as I understand it, is not a doctrine but a theological opinion.
I suspect that the custom of Baptizing infants came about through the work of the Apostles and those who followed in their footsteps, since Scripture was not codified for the first three centuries following Christ's resurrection.  Polycarp indicated he'd been Baptized 86 years (Martyrdom of Polycarp) and Justin Martyr describe those of have been Christian from childhood; these are immediate post-apostolic witnesses.