There is no doubt that, other than unconditional election, limited atonement is the Reformed doctrine of salvation with which people struggle the most. This is usually the case because people don't fully understand how Scripture presents the work of Christ and what it actually accomplished.
The Doctrine Properly Understood
Limited atonement is derived from Scripture which states that the death of Christ actually accomplished salvation for the elect rather than making salvation merely possible or making people merely savable. (Most modern Reformed theologians prefer the term 'definite' rather than 'limited'.) The atonement is limited only to the elect since the death and resurrection of Christ actually saved them from all of their sins when he died and rose again. We see this in a legion of texts of which these are a selection - John 17:2, 9; Romans 5:10; Ephesians 1:4; 5:25, 26; Titus 2:14; 1 Peter 3:18; 1 Thessalonians 5:10; 1 John 2:1, 2.
Let me allow John Owen explain this in more detail with an excerpt from his fantastic book, The Death of Death in the Death of Christ.
“The Father imposed His wrath due unto, and the Son underwent punishment for, either:
1. All the sins of all men
2. All the sins of some men, or
3. Some of the sins of all men.
In which case it may be said:
a. That if the last be true, all men have some sins to answer for, and so none are saved.
b. That if the second be true, then Christ, in their stead suffered for all the sins of all the elect in the whole world, and this is the truth.
c. But if the first be the case, why are not all men free from the punishment due unto their sins?
You answer, ‘Because of unbelief.’ I ask, Is this unbelief a sin, or is it not? If it be, then Christ suffered the punishment due unto it, or He did not. If He did, why must that hinder them more than their other sins for which He died? If He did not, He did not die for all their sins.”
Here is what he means in more modern English.
If Christ actually saved all men (#1) from all of their sins you have universalism; a biblically indefensible doctrine. So we can throw that one out without any further consideration.
If Christ only died to save some of the sins of some men (#3) - unbelief being the sin excluded from this list; then Christ did not accomplish their salvation and thus did not save them. He merely made them savable. This is the Arminian perspective. Let me explain further.
Christ's death makes people savable, but does not accomplish salvation for an individual until they believe of their own libertarian free will. Thus their unbelief, which is clearly a sin, was not covered by Christ's sacrifice. But if this is the case, then how can anyone say that Christ's death was sufficient to save them, if one sin was left out?
Essentially this understanding means that the efficacy of Chris'ts death and resurrection is limited in that it does not save people from all of their sin. This places the Arminian on the horns of a theological dilemma -- "... this unbelief, is it a sin or is it not a sin? If unbelief is not a sin, then why should anyone be punished for it? If unbelief is a sin, then Christ was punished for it in His death. If Christ paid for this sin as all others, then why must this sin stop anyone from entering heaven more than any of the other sins (e.g., murder, adultery, homosexuality, etc.). Furthermore, if Christ did not die for the sin of unbelief, then one cannot say that He died for all the sins of all men. The Arminian cannot escape from the horns of this theological dilemma." (Schwertley)
Thus we are left with only one real option -- If Christ's death actually saved men from their sin (#2), then his death cannot apply to all men. This is the Reformed position. Thus Christ death and resurrection, though sufficient to save all people if God so desired, only actually saves the elect.
Brian Schwertley explains this distinction well.
The Calvinist teaches that Christ’s death is of infinite value to God because Christ was the divine-human mediator. Christ’s death was sufficient to save every man, woman and child who ever lived. In fact, it was sufficient to save everyone on a thousand planets, if God so desired. What limits Christ’s death is that by God’s design and purpose Jesus died only for the elect, those chosen to be saved before the foundation of the world. His death is directed to and actually saves particular persons; not an indefinite mass of people or a hypothetical humanity. Christ offered a definite atonement. It is personal. He knows His own by name (Jn. 10:14).
The Arminian believes that Christ’s death guarantees the actual salvation of not even one person. The Arminian believes in a very limited atonement: an atonement that is weak and impotent to save. God is helpless and waits for the sinner to save himself by choosing Christ. The Father’s plan to save humanity has been defeated, because almost all of mankind has gone to hell. Christ shed His blood and suffered horrible tortures in vain for those who throughout eternity scorn and reject Him. The Holy Spirit has been overpowered and successfully resisted by the vast majority of people throughout history.
The biblical texts that support this doctrine are numerous. The following are simply a selection of them. (For more information and passages see the following sources.)
Matthew 1:21 -- “And she will bring forth a Son, and you shall call His name Jesus, for He will save His people from their sins.” -- There is a great significance in the angels’ expression “His people.” Did Christ come to save every person? Did He come to save the Jews only? No, He came to save His people. “Jesus is not to save every man, but only his own people, for whose ransom he made a pact with the Father, in the covenant of redemption, for it is said, he shall save his own people.”
Matthew 26:28 -- “For this is My blood of the new covenant, which is shed for many for the remission of sins”; Mark 10:45 -- “For even the Son of Man did not come to be served, but to serve, and to give His life a ransom for many.” -- Jesus did not die for all or for just a few, but for the many - the elect.
John 10 -- In verses 11, 14-16 Jesus tells us that the Good Shepherd lays down His life for His SHEEP. Verses 24-30 are even more explicit. In v. 26 Jesus says "... you don't believe , because you are not my sheep." The remaining verses state who ARE His sheep. Notice the personal pronouns: "My sheep hear MY voice and they follow Me ... I give them eternal life, ... neither shall anyone snatch them out of MY hand." Jesus is speaking with the voice of the Son of God, later making Himself EQUAL with the Father.
John 17 -- In his High Priestly prayer, Jesus thanks the Father that he has kept those the Father had given him (v. 12). He makes one exception: "... the son of perdition," referring to Judas. The expression, "son of perdition," suggests that Jesus outwardly called Judas to be one of the twelve, but excluded him from the number of the true disciples. (Cf. Acts 1:25: "Judas by transgression fell, that he might go to his own place.") That place was "perdition" the place of eternal punishment. It was his place from beginning, and Jesus knew it as such (John 12:6). Only the disciples were surprised. Judas was not one of those the Father had given Jesus. In chapter 17 Jesus prays for the eleven true disciples and then (in v. 20 and following) includes ALL who should believe in Him through the word of the Apostles.
Typical Objections to Limited Atonement
Let me deal with only the main texts that are cited against definite atonement. (See the following source for full explanations of these verses; the explanations below are taken from this source.)
1 Corinthians 15:2 - “For as in Adam all die, even so in Christ shall all be made alive.” -- The use of "all" in this passage refers to all in Adam and all in Christ. Adam is the covenant head of all those who die, and Christ is the covenant head of all those who shall live, or all those who will have eternal life. Since all men do not have eternal life, the “all” in Christ cannot refer to the whole human race without exception. The word "all" in the second half of the verse must be restricted to believers. This interpretation is strengthened by the parallel passage in Romans 5:12-21, where it is stated that those in Christ are justified.
2 Corinthians 5:14-15 - “For the love of Christ constrains us, because we judge thus: that if One died for all, then all died; and He died for all, that those who live should live no longer for themselves, but for Him who died for them and rose again.” -- The question is: does all here refer to the whole human race or to the elect - the church? The analogy of Scripture and the context clearly favors the elect only. Paul’s aim in this passage is to motivate Christians to greater obedience by pointing to Christ’s love for us and the judicial union with Christ in His death and resurrection. If Paul was teaching that Christ died for all men without distinction, this passage would prove too much, for Paul’s argument is that this union with Christ in His death and resurrection (which according to the Bible definitely achieves expiation of sin and reconciliation with God) must lead to the service of Jesus Christ - “the love of Christ constrains us.”
2 Timothy 2:3-6 - “For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth. For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all, to be testified in due time.” -- This passage is the one most often cited as the strongest objection to definite atonement. But does it provide such? Not when we understand these verses in context. The context favors translating the Greek word all (pas) as 'all kinds of men'. In 1 Timothy 2:1 Paul says “that supplications, prayers, intercessions, and giving of thanks be made for all men, for kings and all who are in authority.” Paul means that we are to pray for all kinds of people, or all sorts of people—including the civil authorities. Paul’s use of "all" in verse 1 cannot mean all men that have ever existed, or who exist presently, or who shall exist in the future.
2 Peter 3:9 - “The Lord is not slack concerning His promise, as some count slackness, but is longsuffering toward us, not willing that any should perish, but that all should come to repentance.” -- When Peter says that “God is not willing that any should perish, but that all should come to repentance,” who is he referring to when he says all? The word all is clearly restricted by the context to the pronoun us. Peter is clearly referring to believers, to Christians when he says “us” (2 Pet. 1:1). God is not willing that any of us (that is, Christians) should perish, but that all of us (God’s people) should come to repentance. If Peter had meant that God is not willing that any person in the whole world will perish, then this passage would teach universal salvation, for the Bible teaches that God does have the power to carry out His will.
To these we could add John 3:16, 4:42; 1 John 2:2; et. al. The problem that is perceived in these latter passages relates to the word "world." To best understand that this objection is not as simple as it seems we need to understand the ways in which "world" can be used in the New Testament. There are 8 of them:
The word can refer to the entire created order - the universe. (cf. Acts 17:24).
It can refer to the earth itself. (John 13:1; cf. Eph. 1:4).
“World” can mean the evil world system (cf. John. 12:31; 1 John. 5:19).
Sometimes kosmos refers to the whole human race (except Jesus Christ). (cf. Romans 3:19).
Sometimes world refers only to unbelievers. (cf. John 15:18; Revelation 12:9; 13:3; 14:9-10; 1 John. 5:19).
The term world can also be used to describe the Roman empire or what was considered the civilized world in the days of the apostles. (cf. Luke 2:1; Acts 2:5; 19:27; Romans 1:8; Colossians 1:23).
“World” is also used as a synonym for the Gentiles. (Romans 11:12; cf. v. 15, 32).
Sometimes “world” is used as a general term referring to the human race throughout the world. (2 Corinthians 5:19; cf. John 3:16).
What does this all mean? It means that "world" does not necessarily, and most often does not, mean every individual that has ever lived - past, present and future. Context reigns king and we should not allow the meaning of the word "world" to mean more than the context determines.
Conclusion
As R. Scott Clark remarks - "If one accepts that Jesus died as a propitiatory substitute for all his people, there are really only two alternatives, definite atonement or absolute (total) universalism. Either he saved everyone who ever lived, or he saved all those whom he loved."
I also agree with him that - "Scripture teaches that Jesus did not fail. Rather where Adam failed, Jesus succeeded. As the Second Adam (Rom 5:14; 1 Cor 15:22, 44) Jesus actively obeyed God’s perfect Law perfectly, and suffered all the wrath which was due to us, his people, for whom he died (Phil 2:5–11)."
Soli Deo Gloria