Certified GKH Gardening Expert
These are called suckers and often increase when a plant is under stress. Some plants will produce them no matter what.
I suspect that your lime tree is grafted. Check the trunk near the soil like for the grafting collar, which will be a scar the encircles the trunk. many lime trees are grafted to increase their production ability, but a graft does put the tree in a perpetual state of mild stress as the rootstock tries to grow a take over again.
Making sure the tree gets as much TLC as possible will help reduce the suckers but if the tree is grafted, it is unlikely they will go away completely.
It is also important that the suckers be removed. They will be from the rootstock, which will be an inferior or even non-producing variety of citrus. Here is an article that will help you: