We have to investigate whether we are not wanting to see suffering and whether we have given the ability to this person to deal with suffering. - I am not usually in favor of euthanasia.
There is a very inspiring story of a woman who had a terminal illness. She was in a lot of pain and she was very angry, which made the nurses not like her very much, so that when she called for the nurses, she would be the last one that they would got to. But fortunately she had a good meditating friend who was trying to teach her things and one of them, I believe, was the D/D technique. One night she got beyond her own pain and used her pain as a way to open to the human pain of existence and reached a higher level of understanding and compassion, and her personality changed overnight and she became the favorite of the nurses. She was so compassionate and kind.
Human beings have a great capacity to grow through suffering if only there are people who are willing to show them and not encourage them in trying to escape their situation. I saw that in my niece, she had a lot of support from her family, encouragement, and she had an aunt who encouraged that in her.
As far as old animals, it is a bit more difficult. It depends on whether they are domestic or wild. From my experiences of wild animals, they usually go off and fast themselves to death, if they are very ill. So they do it to themselves. For animals - we would like to be able to take care of them as much as possible, however sometimes, because we have taken on the responsibility, it becomes quite difficult to deal with them with some diseases, so it is a much more difficult thing to decide. And that is one of the reasons I don't want to have domesticated animals, because I don't want to decide for them.
For most injured animals, there are veterinarians that can help a great deal to heal an injured animal, rather than just putting it out of its misery. Sometimes this is putting us out of our own misery of opening to Dukkha and not wanting to take the time to try to help that creature.