1. A very weak willed, helpless, or cowardly person; one who lacks any real resolve, inner-strength, or motivation. My father was a sweet man, but he was a limp rag in all other aspects of life, bending to the will of everyone around him. I tried to raise you to be strong and independent, not some limp rag who just coasts through life without ambition. The new supervisor has been as effective at controlling his team as a limp rag.
2. One who is completely worn out or without energy. Between my hectic job and trying to mind the kids, I always feel like a limp dish rag by the time I get to bed at night. My dad used to work crazy long hours when we were kids, and he would just sit on the sofa like a limp rag on the weekends.