In the North particularly many people hunt big game successfully with mouse guns loaded with ordinary bullets; but it is a practice that I would not personally participate in, except in an emergency. A small light TSX can be driven very fast to ensure good penetration even with optimal expansion. This is something the TSX is pretty good at, but you can or rather should only count on 1.5X expansion which would bring the frontal area of a .224 to .336". I suppose the wound cavity from .34 caliber high velocity pill should be enough to result in a clean kill on animals like deer, antelope, and caribou.
GS Custom in South Africa produces mono metal bullets with features that Barnes has copied in some respects. They load a banded light weight .224 bullet on a necked down .30/06 and drive it close to 5000 fps. Their web site shows the carcasses of antelope shot with these things and it does indeed suggest that a small bore can be a successful big game round.
My lightest rifle at the moment is a .243, and I won't even use that for big game. I think a 6.5 is a prudent minimum, but I seem to be a member of a diminishing group with the greater acceptance of new wonder bullets and extremely high velocity. Perhaps Sheephunter is right and I'm stuck in time a generation past. That's OK though, I know the rules and its a comfortable place to be.