Let's make it more personal. If a person has a severly infected arm and the doctor said "we can amputate the arm and you live or you will die if we don't amputate." Anyone would just lop that arm off without much thought.
You have proposed no moral dilemma in your situation. First of all, you have not stated who is making the decision to lop or to not lop. You've also not stated whether the person with the bad arm is capable of making their own decision on the matter. Their
arm is infected in your example, so I see no reason why they wouldn't be able to make their own decision, and therefore you have not proposed a moral dilemma.
This is a moral dilemma:
A person with a severely infected arm is incapacitated and unable to make their own medical decision. The doctor has two possible solutions: he could either amputate the person's arm without consent, thereby saving their life, or he could not amputate and let the person die. Which is the moral decision?
I'll go ahead and answer it. The moral decision, sad to say it, is to do nothing and let the person die.
If the doctor leaves the person alone and lets them die, he is off the hook morally. He did not kill the patient, the infection killed the patient.
But if he takes their arm off without their consent, he has denied them their right to property (in this case a part of their body, and if you don't think one's own body qualifies as their own property then you're out of this world). No matter how appreciative the person is for being alive, or how many Thank You letters they send the doctor, they have been denied that right any way you slice it, no pun intended. In fact, you can plainly see the natural reaction that occurs when somebody realizes they have been duped, in real life as well in movies and whatnot. The, "WTF bro you cut off my arm!" reaction. People
know when their rights have been violated, and they react naturally to such a circumstance even though their life was on the line.
Then society comes along and tells them to forgive the nice man, because society is stupid. Everybody reacts naturally to rights violations, but they don't
understand the rights violations. That's why doctors are forgiven for violating personal property rights every time they amputate a limb or end a life without - or with somebody else's - consent.
Unless of course the patient's Will stated that their permission was previously given for such procedures to take place...