Quote:
Originally Posted by Fordim Hedgethistle
Define, please. 
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To do his duty by seeing his task through to the bitter end. His task, of course, is to destroy the Ring and thereby rid the world of a great evil. There is no hope of reward for Frodo in this ... the likelihood is that he (and Sam) will die even if they are successful.
Nonetheless he undertakes this task, not because there is much chance of success but because it is the only hope the enemies of Sauron have. Therefore it is the right thing to do. It is the duty of all right-thinking people to oppose the encroaching threat of Sauron with whatever means possible. Frodo is the only possible person who can take the Ring to Mordor, consequently doing so is "the right thing" in the sense that it is morally superior to any other option.
Also, in order to keep going on the quest, Frodo has to maintain hope in the face of hopelessness. It is this hope, this faith in the rightness of his quest, that is all he really has to sustain him (other than the support of Sam).