How do I prune this?

How do I prune this?

I am a horrible gardener but I still enjoy having plants etc.... We bought this lime tree last summer(we live in Texas and have mild winters).. and didn't really take care of it.. I noticed it started to die around November or so.. I neglected it even more.. until one day I noticed that it was trying really hard to bloom little leaves.. So, I gave it some plant food and started watering it regularly.. NOW, one side of the tree is thriving and the other side is dead.. I want to cut the dead part off to encourage growth on that side.. but I have NO IDEA how I am supposed to go about it.. would someone love to tell me how I should be doing this? Do I cut all the dead parts off even on the side that is growing? The "main" part of the dead side is still green and alive.. but it has a bunch of dead branches.. Thanks for the help..









Plant ID?

Plant ID?

My new place!

My new place!

Bulb identification

Bulb identification

What Is This?

What Is This?

Flower pickings for today

Flower pickings for today

Habaneros and Amaryllis

Habaneros and Amaryllis