The doctrine of 'Limited Atonement' answers the question of how God saves His elect. He sent His Son to die for them. That's the doctrine plain and simple. Out of all the people that were condemned to hell (everyone) by their own sin, God chose to save some, and then He sent His Son to die for their sins, which He did, and by His own admission pronounced the work of salvation as 'finished'. This is to be understood in light of the doctrine that the 'Arminians' had produced: 'Christ died for all men and for every man, although only believers are saved.'
Again, this is probably the most widely held view of Christ's death in the American Church today. This view says that God is a risk-taker. What greater risk can there be, than, God becoming man and dying for everyone's sins, contingent on their own faith without actually securing the salvation of a predetermined people?
'Arminians' believe that Christ died for everyone, and yet, that not everyone is saved. This calls into question the salvific power of Christ. If Christ died for everyone, then everyone will be saved if we hold to the view that Christ's death was efficacious. If Christ was truly efficacious in His death then everyone must be saved. The only way that 'Arminians' can hold this view is to say that Christ's death was not efficacious in and of itself, but that Christ's death requires our faith to make it efficacious. This makes the 'Arminian' view, in my opinion, weak, and even more troubling, heretical.
Let's look at what Scripture has to say about Christ's death.
Genesis 3:15 "I will put enmity between you and the woman, and between your offspring and her offspring; he shall bruise your head, and you shall bruise his heel."
This verse prophesies of Christ, who would bruise the head of the serpent, while Christ's heel would be bruised. The prophecy is one of victory, contingent not on anything other than Christ's death.
Isaiah 53 1 Who has believed what they heard from us? And to whom has the arm of the Lord been revealed? 2 For he grew up before him like a young plant, and like a root out of dry ground; he had no form or majesty that we should look at him, and no beauty that we should desire him. 3 He was despised and rejected by men; a man of sorrows, and acquainted with grief; and as one from whom men hide their faces he was despised, and we esteemed him not.
4 Surely he has borne our griefs and carried our sorrows;yet we esteemed him stricken, smitten by God, and afflicted. 5 But he was wounded for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his stripes we are healed. 6 All we like sheep have gone astray; we have turned every one to his own way; and the Lord has laid on him the iniquity of us all.
7 He was oppressed, and he was afflicted, yet he opened not his mouth; like a lamb that is led to the slaughter, and like a sheep that before its shearers is silent, so he opened not his mouth. 8 By oppression and judgment he was taken away; and as for his generation, who considered that he was cut off out of the land of the living, stricken for the transgression of my people? 9 And they made his grave with the wicked and with a rich man in his death, although he had done no violence, and there was no deceit in his mouth.
10 Yet it was the will of the Lord to crush him; he has put him to grief; when his soul makes an offering for sin, he shall see his offspring; he shall prolong his days; the will of the Lord shall prosper in his hand. 11 Out of the anguish of his soul he shall see and be satisfied; by his knowledge shall the righteous one, my servant, make many to be accounted righteous, and he shall bear their iniquities. 12 Therefore I will divide him a portion with the many, and he shall divide the spoil with the strong, because he poured out his soul to death and was numbered with the transgressors; yet he bore the sin of many, and makes intercession for the transgressors.
Look at verses 4-6. Notice the occurrences of the words 'we', our, and us.
In verse 8 He died For the transessions of 'my people'.
In verse 10 the idea of a definite atonement is shown by Christ 'seeing his offspring' while He is making an 'offering for sin'.
In verse 11 He is satisfied by His accomplishment of His anguish.
In verse 12 He bears the sins of many (not all).
Matthew 1:21 "She will bear a son, and you shall call his name Jesus, for he will save his people from their sins."
This is part of Jesus' birth announcement. The angel tells Joseph that Jesus "will save His people from their sins."
John 10:11 "I am the good shepherd. The good shepherd lays down his life for the sheep."
Notice here that Jesus States plainly that He dies for the sheep.
2 Corinthians 5:14-15 For the love of Christ controls us, because we have concluded this: that one has died for all, therefore all have died; and he died for all, that those who live might no longer live for themselves but for him who for their sake died and was raised.
O. Palmer Robertson in an essay on 'Definite Atonement' states,
"At first reading, it might appear that this statement affirms that Christ's death was effective for all people. But it must be noted that in this statement 'all' those for whom Christ died are 'all' those who themselves have died. Who then are the 'all' who have died? This dying by all is obviously not referring to physical death, since many people have not yet died in this sense. Instead, this dying refers to a dying to self, as the next verse explains: "He died for all, that those who live should no longer live for themselves but for him who died for them, and was raised again. (v. 15) In this verse, the expression "those who...no longer live for themselves" clearly does not refer to all human beings, but to all those who have been made alive in Christ. These are the 'all' for whom He died, that is, all those who are in Him."
(Taken from: After Darkness, Light; Essays In Honor Of R.C. Sproul. Edited by: R.C. Sproul Jr. Presbyterian & Reformed Publishing Co. Box 817, Phillipsburg, N.J. 08865. 2003. Pages 104-105.)
Titus 2:11-14 11 For the grace of God has appeared, bringing salvation for all people, 12 training us to renounce ungodliness and worldly passions, and to live self-controlled, upright, and godly lives in the present age, 13 waiting for our blessed hope, the appearing of the glory of our great God and Savior Jesus Christ, 14 who gave himself for us to redeem us from all lawlessness and to purify for himself a people for his own possession who are zealous for good works.
These verses, especially verse 11, are used by the 'Arminians' to say that God's grace has appeared to every individual person by virtue of Christ dying for every individual person. In context the verse does not refer to individuals, but to classes. Paul speaks of Titus, older men, older women, (sorry), young women, husbands, wives, children, young men, masters, and slaves. Why? Because the grace of God has appeared, bringing salvation for all people (classes). Following on, this grace teaches us how we should live, because Christ gave Himself for us, that we should be His own people.
'Limited Atonement' means limited in scope, not in power or sufficiency, as some have presented the doctrine. I am not one for changing terms in order to make them more palatable, but 'Definite Atonement' might be a better description of the doctrine in this regard. Christ died for a definite people, had a definite purpose, and definitely succeeded in that purpose.
Had Christ died for every individual in the world, He would have succeeded in saving every individual in the world. The fact that not every individual in the world is saved demonstrates the fact that He did not die for every individual, but for the elect.