[quote=DavidFilmer]That seems logical, but the problem here is that a person on his deathbed may not have the time or the stamina to profess such an understanding. It might be the case that the best he can do is request to be admitted to the Final Sacraments. Presumably the priest has never seen this person before. What is the priest to do?
well, baptism is the sacrament that is necessary for salvation. (Of course, this is not to say that God will not save someone who has not, out of ignorance of the sacrament, been baptized.) The priest would definitely baptize an unbaptized person, and any person in fact could baptize an unbaptized person in the name of the Father, Son and Holy Spirit.
As for the other sacraments, it’s not like a priest carries around the Eucharist with him. If a person is dying and there is time to get the Eucharist to him, there is time for the person to make a profession of faith or at least express to those around him his intentions. The person must make it explicitly clear that he believes what the Church teaches in order to receive the other sacraments. I think a lot of times, though, much is left up to the priest’s judgment.