the lord of the rings - Why did they take a Palantir with them when they left on the last ship going west to the Undying Lands?

First, to explain some backstory of the Palantiri.

The Palantíri were made by the Ñoldor in Eldamar, possibly by Fëanor himself in Aman during the Elder Days in the Time of the Trees, and then given by the Elves to the Númenóreans, who kept them as heirlooms until the Fall of Númenor during the Second Age. Then seven of these stones were rescued and brought to Middle-earth by Elendil and his sons.

The Dúnedain placed the stones across large distances in order to communicate with one another.

The stones were housed at the following locations: Annúminas, Amon Sûl (Weathertop) and Elostirion (Tower Hills) in the north, and Osgiliath, Orthanc (Isengard), Minas Ithil, and Minas Anor in the south. The Master Stone resides in Tol Eressëa, in the Tower of Avallonë.

Four of the stones are known to have been lost. The chief stone of the north at Amon Sul, along with the stone from Annúminas was lost with Arvedui in the cold northern seas. The chief stone of the south in Osgiliath was lost during the Kin-strife. The stone of Minas Ithil was captured by Sauron and was very likely destroyed during the destruction of Barad-dûr.

The stone of Elostirion was taken back to the Undying Lands (Valinor) on the Ringbearers' ship. Only two stones remain in Middle-earth; the stones of Minas Anor and Orthanc, yet the stone of Minas Anor was marred, showing all but the most strong-willed the sorrows and madness of Denethor II.

Back in the day, the Palantiri were most readily available to heirs or kings, as well as those appointed to guard them. The stones were able to sense their user and allow use based on their position; hence Denethor could utilize the stone of the White City easily, whereas Saruman struggled with its use, and was eventually overcome by Sauron. Palantíri, wikia

So to the question proper; The Palantir that Saruman had in Orthanc was in due course delivered to Aragorn as was stone Denethor had in Minas Tirith (Minas Anor). That leaves the reigning King in middle earth with two of the three surviving Palantiri and the third taken across the sea to the Undying lands. Why was the Palantir of Elostirion taken back to the Undying Lands? Would this mean that Aragorn had ongoing communications with Valinor?

Answer

Short answer:

The Palantír of Elostirion was designed to only look west towards the Master-stone in the Tower of Avallonë and didn't communicate with the other Palantíri in Middle-Earth. It was likely removed from Middle-Earth to preclude any possible 'communication' or 'viewing' of the West by those left in Middle-Earth.

Long answer:

The Palantír of Elostirion, also known as the Elendil Stone was one of the Palantíri brought by the Faithful from Númenor. It was stored in the tower Elostirion on the Emyn Beraid by Elendil, and in later years was guarded by Círdan and the Elves of Lindon.[source?]

The stone was aligned westwards towards the Master-stone in the Tower of Avallonë, along the length of the Straight Road; because of this, it could not communicate with the other six of Middle-earth. Only Elendil was able to use it to look west across the Sea and see the Undying Lands;[1] Elendil also tried to see the fallen Númenor with the stone but failed.

Even after the fall of Arnor, Wandering Companies of Elves made pilgrimages to visit the Towers and see the Stone, in order to catch a glimpse of Varda. One such company, led by Gildor Inglorion just returning from Elostirion, was encountered by Frodo, Sam and Pippin on September 24, T.A. 3018.[2]

It remained in its tower until the end of the Third Age; it was then taken back into the West aboard the White Ship.[3]

References 1.↑ J.R.R. Tolkien, Christopher Tolkien (ed.), The Silmarillion, "Of the Rings of Power and the Third Age" 2.↑ J.R.R. Tolkien, The Lord of the Rings, The Fellowship of the Ring, "Three is Company" 3.↑ J.R.R. Tolkien, The Lord of the Rings, The Return of the King, "The Grey Havens"

Elostirion was the tallest and westernmost of the three White Towers that stood on the Tower Hills[1] and that had been built by Gil-galad for Elendil. It held one of the palantír of Arnor which did not communicate with the rest but looked only westward across the Sundering Seas to Tol Eressëa. This, the last palantír of Arnor, was carried back across the sea with the departure of the ring-bearers early in the Fourth Age.[2]

References 1.↑ J.R.R. Tolkien, The Lord of the Rings, "Prologue", "Concerning Hobbits" 2.↑ J.R.R. Tolkien, Christopher Tolkien (ed.), Unfinished Tales, "The Palantíri" http://tolkiengateway.net/wiki/Elostirion

Note: This is the best explanation I could find and will accept it unless someone can better explain.