I WON, I thought you might have said Mexico, as some say Limes did come from that area, a few might have said the West Indies, as Jamaica is a vast grower, however think you will find the English brought the Lime Cuttings to start the growing in that area, so that our Navy could take them on board to give to the crew as a defence against scurvy, hence the term Limeys.
Why I said China, known ancient pottery from China shows Lemons, Limes with the Cherry Blossoms on them, could have been Limes went to China from India with Buddha, one will never know.