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

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.

The elves are immortal by nature and it isn't Valinor that makes them immortal (they would have lived forever in Middle Earth, too, they just go to a nicer place to live). Likewise does it not cause the mortal hobbits to live forever. It is called the Undying Lands because only immortals live there and not the other way around.

But on the other hand, this Wikipedia site says:

...only immortal beings were allowed to reside there, but the land itself, while blessed, did not cause mortals to live forever. Amongst the exceptions to this were the surviving bearers of the One Ring — Bilbo and Frodo Baggins and...

But the phrasing here is a bit unclear. It could be they were an exception to only immortals being allowed there, or they could be an exception to the fact that it didn't cause mortals to live forever.

It's been quite a long time since I read LotR and The Silmarillion, but I don't think Frodo and Bilbo were going to live forever. They just got a nice place to die.

No, The phrase "Undying Lands" is not a place of immortality bestowed. It was a realm inhabited by Ainur (spiritual beings) and Elves whom are already immortal. The "Undying Lands" do not grant immortality, the Valar do not have the power; OR the authority to bestow immortality upon mortals (though they did "bless" the Men loyal to them during the War of the Wrath, the "Numenoreans", with extended lifespans).

The Undying Lands are simply named for the immortal inhabitants. The Ringbearers travel there to receive healing for the wounds (spiritual or physical) incurred as a result of ownership of the One Ring; the effects of the Ring is ubiquitous (albeit differing in time) to all beings immortal or otherwise. They would still die, but the hope was that they could live the rest of the time without Suffering or if they could be healed; allowed to return. The elves that bore rings of power also went there after the power of their rings was "spent" thus their domains would deteriorate.

In Tolkiens letters he describes... Frodo was sent or allowed to pass over Sea to heal him - if that could be done, before he died. He would have eventually to 'pass away': no mortal could, or can, abide for ever on earth, or within Time.

—Letter 246

Also stuff in Valinor happens very... very slowly since all its usual inhabitants are immortal there's little need to rush tasks, and 100 years would seem like just a mere blink of an eye, so really its not a nice place for a mortal person to reside; less they're left to rest.