What does Eli, Eli, Lama Sabachthani mean in the Bible?

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Holman Bible Dictionary - Eli, Eli, Lama Sabachthani
This cry of Jesus on the cross, traditionally known as the “fourth word from the cross” means, “My God, my God, why hast thou forsaken me?” (Matthew 27:46 ; Mark 15:34 ). It is a quotation from Psalm 22:1 . The Markan form, Eloi , is closer to Aramaic than Matthew's more Hebraic Eli . Both the evangelists translated the quotation into Greek for their readers.
Scholars are uncertain whether Jesus spoke these words in Hebrew or Aramaic. If He spoke in Hebrew it would more readily explain the crowd's confusion of “my God” with “Elijah,” since the terms sound more nearly alike in Hebrew than in Aramaic.
This saying of Jesus from the cross strikes a dissonant chord for some Christians, because it seems to indicate that Jesus felt forsaken by the Father. There are several ways to consider the meaning of this passage in reverent faith. It is possible to interpret these words as a beautiful testimony to Jesus' love of His Bible, the Old Testament, and His quoting of it in this hour of darkest crisis. In this case, such verses in Psalm 22:1 as 5,7, 8,12, 14, and 18 indicate that Jesus sees Himself and His fate in this Psalm. However, since the Gospels record only the first verse of the Psalm and we do not know whether Jesus quoted the entire Psalm, this view may run the risk of not taking the phrase at face value.
Another view sees this cry as indicating a genuine forsaking of Jesus by the Father, a forsaking which was necessary for our redemption. This view leads some to questions about the nature of the godhead and theories of atonement which we cannot address in this brief discussion. Perhaps the most serious difficulty of this view is that it raises the question of whether the idea of God the Father turning His back on the obedient Son is consistent with the general biblical teaching of the steadfastness and faithfulness of God. Would He desert a trusting child in such an hour?
A view which takes into consideration the full humanity—as well as full divinity—of Jesus seems most helpful. Obviously Jesus felt deserted as He bore the burden of human sin and suffered the agony of crucifixion. This feeling of His death as a “ransom for many” may, indeed, have obscured for a time His feeling of closeness with the Father, so that even in dying He was tempted as we are. Rather than forsaking the Father in that moment, He cried out to Him in prayer.
Earl Dans
Hastings' Dictionary of the Bible - Eli, Eli, Lama Sabachthani
ELI, ELI, LAMA SABACHTHANI . See Eloi, Eloi, etc.
The Hawker's Poor Man's Concordance And Dictionary - Eli Eli Lama Sabachthani
The reader will not wish to pass over this well known cry of Jesus on the cross; but will be gratified with the continued attention of it. Those words of Christ are full of important signification; and every pious reader of his Bible ought to have a proper conception of their meaning. They are partly in the Hebrew, and partly in the Syriac tongue, and which, perhaps occasioned the perverse misconstruction in some, who supposed the Lord called Elias, when Jesus said Eli. The prophet had said, "That the Lord should roar out of Zion, and utter his voice from Jerusalem, and the heavens and the earth should shake." (Joel 3:16) And hence we find that prophecy fulfilled. The loud voice of Jesus was not like one whose strength was gone, but rather uttered in proof of what Jesus had said: "No man taketh my life from me, but I lay it down of myself, I have power to lay it down, and I have power to take it again. This commandment have I received of my Father." (John 10:18) The words themselves seem to be a quotation from Psalms 22:1, thereby intimating, that the prophet in that Psalm spake wholly of Christ. This was highly important for the church to know. And the meaning yet more important. The Holy Ghost hath caused his servants the Evangelists, to give the church the interpretation: Eli, Eli, lama, are Hebrew; Sabacthani, or Sabadetani, is Syriac. Astonishing words for the only beloved of the Father to utter! Jesus had uttered no cry of pain in the great tortures of his body; neither do we hear the meek Lamb of God complain of the insults of the rabble, in the unequalled repreaches cast upon him. These, and every other sorrow, seem to have been swallowed up and forgotten in the flood of divine wrath, which now opened like cataracts from heaven in the Father's desertion. Who shall say what this was? Who is competent to describe the horrors of it, when it induced such a cry in the soul agonies of Jesus? Well may every child of God pause over the renewed reading of it, and in the contemplation, consider the love and tenderness of Jesus to his people, who thus endured the being forsaken of his Father for a season, that they might not be forsaken for ever. (Hebrews 5:7-9)

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Eli, Eli, Lama Sabachthani - Eli, Eli, Lama Sabachthani
Languages of the Bible - New Testament Greek is heavily infused with Semitic thought modes, and many Aramaic words are found rendered with Greek letters (for example, talitha cumi , Mark 5:41 ; ephphatha , Mark 7:34 ; Eli, Eli, Lama Sabachthani , Mark 15:34 ; marana-tha , 1 Corinthians 16:22 )
John, Gospel of (ii. Contents) - ), following the miracle of the 5000, and does not wish the truth of the mystical union to be bound up too closely with the participation in an ecclesiastical rite; he omits the Agony in the Garden of Gethsemane, and the cry, ‘Eli, Eli, lama sabachthani,’ because the impression which he wishes to convey of the complete voluntariness of Christ’s sufferings and death, and of the ‘glory’ which was manifested by His humiliation as well as by His triumph over death, might be impaired by incidents which seem to indicate human weakness and hesitation; and, lastly, he omits the Ascension and the descent of the Paraclete, because he does not wish the withdrawal of Christ’s bodily presence, and the continuation of the Incarnation in another more spiritual form, to be associated with physical portents, or to be assigned to particular days