You have a citrus tree growing in your fence. The new lime tree isn't far away. I would move it at least 10 ft away. The other tree has larger roots and is going to take all the nutrients. It is not growing in a good place and really should be removed.
When did you plant the tree?
If it was recently planted, did you bury it at the same level or lower. No tree should be buried lower than the soil level in the pot or the stem will rot and the branches will die.
https://ucanr.edu/sites/VCMG/Planting_an ... rus_Trees/
Since your other tree has been growing there for awhile, you have to be in a zone that is good for the tree. It is better if you update your profile with your zone and location since your local conditions will determine what is going to work.
Usually citrus trees are fertilized only when they are in a growth phase.
If the tree was properly planted and the roots and stems are not rotting, then you need to seed what else could be the problem.
Do you see ants, aphids or scale on the plant leaves, stems.
The tree should be watered and the soil should be well drained and not puddling. Citrus trees like to be almost dry but will not tolerate sogginess. If it is not getting enough water the leaves will first curl and dry before they drop off. If it is getting too much water, the leaves will yellow and drop.
Check if the leafless limb is still alive. Is it pliable or brittle. If it is brittle it is already dead and needs to be cut back.
It could be transplant shock, it could be over watering or some issue with the roots not being able to transport nutrients to the top of the plant. Either too much or too little water, poor drainage, rotting roots, poor soil.
Since I think the plant needs to move further away anyway, dig it up and make sure the roots are still in good condition and they are alive.
Dig your new hole in a sunny spot that is well drained and not near any other tree roots or near the house. Citrus trees will grow to 30 ft tall at maturity so you want them away from walls, house foundations or pipes. Do not plant them too deep. Back fill with good soil. If you have to fertilize, use only slow release fertilizer. Water in well and water so that the soil is moist but not soggy. Put out ant bait to control the aphids and scale. Feed citrus when you see new leaf growth.