Andy, my best guess from the quality of the picture would be a cat palm. A close up of the trunk would help. If it is a cat palm I do not anticipate you would have much problems with the conditions you describe.

Ed- Thank you for the information. It didn't look so dark through the camera when I took the picture. If it's a cat palm is it possible that a new trunk will form. Like I said i had to cut one of the foure trunks off beacause it was going limp. There does seem to be tiny little green growths coming out of the main trunk about two inches long).

I believe that cat palms grow offshoots, so I don't think it is a problem that you cut off one of the trunks. A new one might simply replace it. They are pretty common in the big box stores in Michigan and are fairly good palms to grow idoors.

That's not a cat palm, it looks like a kentia to me. A better picture of the 'trunk' would help.

Flound-
I wish the picture had come out better. I know it's not a Kentia, I know what those look like. It does not have the "burlap" groing around the trunks. In adiition Kentias are too valuable up here for one to be left to die in a grocery store parking lot. One would only find them in higher end nurseries up here. I'll try and get a better picture and re-post it.

It's a chamaedoreum cataractum (cat palm).

Andy I agree it is a cat palm I have them in my nursery it will grow fine in a pot inside but don't move it around too much