Front doorway concrete cracked

Our 1961 Atrium Eichler in Terra Linda has a fairly nasty crack right outside of the door, running parallel to the door. The outside piece dips slightly downward from the door. And it just didn't break away, it buckled: there's a height difference to the inside. That is, the outside piece is about 3/4" higher than the interior rest of slab where they meet.

Question: Is this concrete area in the doorway attached to the slab of the house? Or is there a foundation in-between?

I'm thinking of tiling the atrium, starting in this area. One idea is to take out (jackhammer, pick, etc.) this concrete piece and repour. But would I jeapardize the slab under the house and the adjacent garage? If it's separate, it shouldn't be a problem. It's clearly connected to the atrium slab (if it weren't cracked).

Of course, the other obvious choice is to rent a grinder, make the crack area even, tile it and hope it doesn't crack more.

Ideas? Thanks for any input.

--Bob