To propagate a lime tree, the most common method is by taking a semi-hardwood stem cutting from a healthy branch, treating the cut end with rooting hormone, and planting it in a moist, well-draining potting mix; you can also propagate lime trees from seed, but this method takes longer and may not produce fruit identical to the parent plant; for faster results, professionals often use budding or grafting techniques to propagate lime trees onto established rootstocks. 

Key points about propagating lime trees:

Cutting selection: Choose a healthy, semi-hardwood stem cutting around 4-6 inches long with several leaf nodes. 

Rooting hormone: Dip the cut end of the stem in rooting hormone to stimulate root development. 

Potting medium: Use a well-draining potting mix with good aeration. 

Environment: Place the cutting in a warm, humid environment with bright, indirect light. 

Other propagation methods:

Seed propagation: Extract seeds from a ripe lime. 

Rinse off the pulp and plant the seeds in a seed starting mix.  Expect longer time to fruit compared to cuttings. 

Air layering: Make a ring-shaped cut around a branch on the parent plant. 

Wrap the cut area with moist sphagnum moss and cover with plastic to encourage root development. 

Grafting/Budding: Used by professionals to propagate specific lime varieties onto established rootstocks.