In most cases, yes, however, in most stories, those blessed with immortality usually resent the fact that they're unable to die and jump at the chance to be able to, mostly because they see so many generations come and go, see their friends and family die, and be unable to do anything about it, as well as be unable to follow them into death, where they could at least be together. Though 8 billion years from now I might rethink the fact of becoming immortal. At least I can't die when the sun goes out and the temperature drops to absolute zero.