At what point is a picked flower considered dead?
This is very much analogous to your "limb" suggestion, or, since the flower is a reproductive organ, one can ask, 


  At exactly what point is the uterus considered dead after a hysterectomy?


or,


  At exactly what point are the testicles considered dead after a male dog is castrated?


The organism (woman and dog respectively) does not die. Tissue dies. Tissue dies all the time. Red Blood cells die after about 120 days, give or take. Skin cells die faster than that.

It depends on the definition of death. If one of the requirements for "life" is the ability to reproduce, are both the woman and the dog now dead? Of course not. They had the potential, or perhaps they already reproduced.

If one of the requirements for "life" is the ability to grow, well, the uterus and the testicles are dead immediately upon removal. But we stop "growing", and we don't die.

If one of the requirements for "life" is the ability to respond to its environment, the uterus and the testicles are not dead. They are "alive" for a time, though the environment is toxic to them if thrown away. If placed in nutritious media, they will "live" a bit longer.

Common sense, which often isn't reliable in science, would dictate that the flower dies when it can no longer maintain cell turgor, i.e. can no longer absorb fluids. But the argument can be made that it is only "dying" at that point. 

Basically your question depends entirely on the definition of life. And that has not been agreed upon.