Why is grout at tub edge disintegrating?
It  is  probably failing because grout was in the gap prior to you caulking over it and it was not clean enough.
The gap should not have been grouted at all. You should remove all of the caulk and grout, use a  retractable utility knife and a five in one tool to clean it out. Use the flexibility of the knife to get as much of the old caulk out of the gap and off of the tub lip. (protect the tub lip with some painter tape) Clean the gap well with some denatured alcohol and 3M scouring pad and dry with a towel. (It is vital that you get all of the old caulk off.)
It should be filled with 100% silicone, mold/mildew resistant, caulk.
Tape off the tile and the tub with painters tape so that just the thin gap that you want caulk to be is exposed and then apply the caulk with caulk gun.
There are many tutorial videos on YouTube that will give you instruction on this technique; How to caulk a tub YouTube search.
Here are two
Ask this old house.
See Jane drill.