Why did Frodo go to the Undying Lands in the end?

The only way for Frodo to find peace was for him to leave Middle-earth and head to the Undying Lands, in the hopes that this mystical realm could finally heal his injuries forever.

Is Frodo immortal in Undying Lands?

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.