Juice the limes, and keep aside in a large pitcher.
Using a food processor or blender, grate the carrots.
Switch the attachment to blender/puree, add some water to the carrots, and pulse briefly.
Working in batches, push the carrot mixture through a sieve so that the juices run out and the solids are left behind. Add the juice to the pitcher, and keep the carrot solids aside. Continue with the remaining carrot mixture until you've fully separated the solids and juice.
Return the solids to the food processor, add a bit more water, and repeat the process to extract more juice.
Stir sugar or simple syrup into the juice mixture, taste, and adjust with sugar or lime if desired.*
Serve over ice.
Notes
* Experiment with the lime and sugar per your taste, sweeter or more tart. There are as many recipes for this as there are chefs