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and as the land is Undying, it is probable that there they are immortal.
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Gwaihir, the reason Aman was called the Undying Lands was because of the people that lived there, not because of any special properties that the land itself possessed. In
Unfinished Tales, the Elves told the King of Numenor that living in the Undying Lands did not confer immortality, rather, it was the people themselves who were immortal.
Therefore, Shadowfax wouldn't have immediately become "immortal" just by living in the Undying Lands.