Edwin Palmer (d. 1980) wrote a helpful book called The Five Points of Calvinism. This is a good resource for those who want a detailed yet readable and relatively brief explanation of the doctrines of grace. Here’s an edited summary of Palmer’s discussion of Limited (Definite) Atonement which he explains in a Trinitarian and biblical way. The doctrine of Limited Atonement is based on:
1) The Father’s Election. Since the objects of the Father’s saving love are particular, definite, and limited (Amos 3:2, Rom. 1:7, 8:29, 9:13, Col. 3:12, 1 Thess. 1:4, Jude 1) so are the objects of Christ’s death. Because God has loved certain ones and not all, because he has sovereignly and immutably determined that these particular ones will be saved, he sent his Son to die for them, to save them, and not all the world. Because there is a definite election, there is a definite atonement. Because there is a particular election, there is a particular atonement. God’s electing love and Christ’s atonement go hand in hand and have the same people in view. There is unity between the work of the Father and the Son.
2) The Son’s Atonement. The Bible teaches the death of Jesus in at least four different ways. When Christ died, 1) he made a substitutionary sacrifice for sins (Heb. 9-10); 2) he propitiated, that is, appeased or placated, the righteous wrath of God (Rom. 3:25; Heb. 2:17, 2 John 2:2; 4:10); 3) he reconciled his people to God – that is, he removed the enmity between them and God (Rom. 5:10, 2 Cor. 5:20, etc.); and 4) he redeemed them from the curse of the law (Gal. 3:13). …The nature of the atonement – what did Christ actually do? – answers the question: For whom did Christ die? The noun (atonement) defines its adjective (limited). If the atonement does not actually save, does not really remove God’s curse from people, does not actually redeem them, then it indeed can be for all the world, even for those who are in hell. But if the death of Jesus is what the Bible says it is – a substitutionary sacrifice for sins, an actual and not a hypothetical redemption, whereby the sinner is really reconciled to God – then obviously, it cannot be for every man in the world. For then everybody would be saved, and obviously they are not.
3) The Spirit’s Indwelling. In 2 Corinthians 5:14-15, Paul notes (in line with Romans 6) that if Christians are dead to sin, then they are made alive in Christ. If they are spiritually buried with Christ, they will spiritually rise with him. Although Paul does not state it explicitly in this passage, we know from the rest of Scripture that this is possible only through the Holy Spirit’s work. …There is an inexorable chain of events in 2 Corinthians 5:14-15: a) Christ died for all believers; therefore b) all believers die spiritually in Christ; and c) they all rise again spiritually in Christ. If (a) is stated, (b) and (c) must follow. …The Holy Spirit does not apply the death of Christ to all people, leaving it in their hands ultimately as to whether or not they would be saved. Rather, the Spirit comes to those people whom the Father had chosen and for whom the Son had died and he causes them to die to sin and be born again.
In summary, the purpose of the Father, Son, and Holy Spirit coincide. They strive for and accomplish the same purpose: The salvation of those whom the Father has loved with a special love.
To read these three points in their entirety, see pages 52-60 in The Five Points of Calvinism (Grand Rapids: Baker, 2010).