Passages which seem to contradict a limited view of the atonement do not actually contradict it. Some of these passages do just the opposite.
For example, consider the classic proof text for a general atonement: John 3:16. This passage teaches that God gave Christ to the world so that believers might be saved. All believers are elect and all the elect eventually believe. Therefore, even John 3:16 teaches a limited intention for sending Christ into the world—to save the elect (i.e. all who believe). One does not need to interpret “for God so loved the world,” to mean “for God so loved the elect,” for this to hold true. The purpose clause “so that” is limited to the elect regardless of how broad the scope of meaning for the term “world.” God loves the world, but he sends his son for the benifit only of whosoever believes, and whosoever believes is elect.
“For God so loved the world, that he gave his one and only Son, so that whosoever believes in him would have everlasting life.”
John Owen notes that the meaning and usage of those terms which are universal in form—such as “world” and “all”—must be weighed very carefully for this reason: “Upon these expressions hangs the whole weight of the opposite cause, the chief if not the only argument for the universality of redemption.”  Once the full range of meaning for these words is closely examined, however, the biblical objections to limited atonement are less convincing. The word “world” (kosmos) in Scripture does not always refer to every person in the world without exception. There are many passages where kosmos simply cannot mean every individual human being (Jn 7:7; Rom 1:8; 1 Cor 4:9; 11:32). If one is to believe that Christ died for everyone without exception on the grounds that the Bible says he died for the sins of the kosmos, she unwittingly gives good reason to think that everyone alive in the first century was a follower of Jesus, since the Pharisees exclaimed, “Look, the world [ho kosmos] has gone after Him” (Jn 12:19).
Even more important, kosmos often refers only to those who believe. For example, Paul taught that Israel’s sin of rejecting Christ means “riches for the world” (ploutos kosmou, Rom 11:12). Can we say then, that every person in the world without exception has received the ploutos Paul has in mind? It seems clear that Paul is using the word “world” to distinguish between Jew and Gentile, and that he would intend us to understand only those who believe in Christ as the recipients of the riches Paul has in mind in this context. Such an interpretation, however, leads us to conclude that kosmos actually refers to a minority group among the people in the world—the few that find the ploutos in Christ (i.e. the elect).
When the apostle John admonishes his readers not to think of Christ’s death as for them only but for the “whole world” (1 Jn 2:2), the grammatical structure is strikingly similar to statements found in his gospel (Jn 11:51-52). On the basis of this parallel one might conclude that “whole world” in his epistle simply refers to God’s people, the elect, scattered throughout the whole world.
Although many passages describe the death of Christ as being for “all” (pas, Rom 5:18; 1 Cor 15:22; 2 Cor 5:14-15; 1 Tim 2:4-6; Heb 2:9; 2 Pt 3:9), like the word “world,” the word “all” in Scripture does not always refer to everyone, but it must be determined by context. Sometimes the word “all” simply refers to all those within a certain group defined by the context. For example, Romans 5:18 teaches that just as one sins led to condemnation for “all,” so one act of righteousness results in justification for “all.” Here, even within the very same context, one must interpret the former reference to “all” as virtually universal, and the latter as limited only to believers. Without allowing for such fair distinctions based on context, the interpreter has no way to object to the conclusion that all people without exception are justified before God. Paul’s statement in 1 Tim 2:6 that Christ was given as a ransom for all can simply mean “all kinds,” (indiscriminately with respect to Jew, Gentile, male, female, slave, free). In fact, Paul’s usage of the word “all” is best understood this way based on the way he uses it in the context (cf. 1 Tim 2:1-2). “All” in Titus 2:11 can be taken in a similar way based on context (cf. Tit 2:2-4, 6, 9).
 John Owen, The Death of Death in the Death of Christ (Carlisle, Pennsylvania: Banner of Truth Trust, reprinted 1999), 190.
 I owe this insight to John Piper. John Piper, Tulip: The Pursuit of God’s Glory in Salvation (Minneapolis, Minnesota: Desiring God Ministries), 31.