plot explanation - Do Frodo and Bilbo live forever at the end of the Lord of the Rings trilogy?

Elrond mentions that going to The Undying Land which is west of Middle Earth allows Elves to be immortal.

Does this apply to other creatures such as Hobbits?

I know Frodo and Bilbo Baggins go there at the end of the Lord of the Rings but do they go there to die or to be immortal?

Answer

I am delivering here a relevant portion of a forum thread concerning the immortality of mortals who pass to the Undying Lands. It seems that Christopher Tolkien used many of his father's letters to accumulate a fair body of knowledge about Middle Earth and its rules in The Silmarillion, but that some letters specific to the life and death of Frodo and Bilbo Baggins were unclear, (perhaps intentionally so - my interpretation - The whole thread is a very interesting read).

Here is the relevant passage I found:

But in this story it is supposed that there may be certain rare exceptions or accommodations (legitimately supposed? there always seem to be exceptions); and so certain 'mortals', who have played some great part in Elvish affairs, may pass with the Elves to Elvenhome. Thus Frodo (by the express gift of Arwen) and Bilbo, and eventually Sam (as adumbrated by Frodo); and as a unique exception Gimli the Dwarf, as a friend of Legolas and 'servant' of Galadriel.

I have said nothing about it in this book, but the mythical idea underlying is that for mortals, since their 'kind' cannot be changed for ever, this is strictly only a temporary reward: a healing and redress of suffering. They cannot abide for ever, and though they cannot return to mortal earth, they can and will 'die' - of free will, and leave the world. - Letter #154

Or from letter #325:

As for Frodo or other mortals, they could only dwell in Aman for a limited time - whether brief or long. The Valar had neither the power nor the right to confer "immortality" upon them. Their sojorn was a "purgatory", but one of peace and healing and they would eventually pass away (die at their own desire and of free will) to destinations of which the Elves knew nothing.