How to repair a broken patio corner?
I am really not sure what your patio looks like.  I am going to assume it sits on the ground.

It would be normal for a lot of the erosion of the patio to happen on the corners.  You have basically two options...


Compress ground.  You need to wet the ground around the area to make sure there are no gaps with weight.  The pavers shouldn't be hanging above ground at all.  If this is the case then you will have more that breaks.  An easy fix would be to get some fast setting quickcrete...  Build yourself a box to meet your needs - you will probably need 2 pieces of 2x6.  Fill box with the quickcrete and try to push the quickcrete under the adjacent paver piece if there is a gap.  Just level it out to fit your needs.  Pour slowly so you don't have to cup any out.
Remove the current piece.  Keep removing pieces until entire paver lays very flat on the ground with absolutely no gapping.  Add more dirt/rock and then compress and reinstall pavers.
Based on picture I would carefully knock off as much of the cement as possible inside the boxed corner - if that makes sense.  Really I don't understand the lip.  Maybe it looks good but will increase chance of stone cracking.  I would then refill with a quick set concrete.  It needs to be filled right.  You see yours is uneven probably helping the break.  I would then put that corner back on... I know you will have a little crack showing but taking a sheet out will be a pain.  I would then fill the crack with a knife grade epoxy - should be fine with limestone.  Since your color is lighter and very neutral I would not color the epoxy or it might stand out more.