1 John 2:2 - The Silver Bullet?

 Is 1 John 2:2 the fatal blow to the doctrine of Limited Atonement? 

1 John 2:2 "and He Himself is the propitiation for our sins, and not for ours only, but also for those of the whole world." (LSB)

Limited Atonement, the L in the TULIP acronym (which is descriptive of the Calvinists' view of soteriology), is the idea that Jesus Christ did not die for the whole world, but only for the elect. Now at first glance, this passage seems to rule out any possibility of that being true. But I think that is often the issue whenever this text is used by someone who rejects Calvinism. It may not be true in every case, but more often than not, they never go any further than the first glance at this text. They see the use of "whole world" and say "ah ha! It says whole world, that means He died for every single person in the world!". So let's take a closer look and see if this text is indeed the silver bullet to the doctrine of Limited Atonement. To do so, I'd like to add the verse prior to our chosen text to add a bit of context.

1 John 2:1-2 "My little children, I am writing these things to you so that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous; and He Himself is the propitiation for our sins, and not for ours only, but also for those of the whole world."
I want to make two quick and simple observations to show that verse 2 does not refute the Calvinist's doctrine of Limited Atonement:
1. Verse 2 is a continuation of verse 1. They are directly linked together. In verse 1 we see that if we sin, Jesus Christ is our Advocate with the Father. According to the Manual Greek Lexicon of the New Testament, advocate, παράκλητος (paraklētos), means "called to one's aid in in a judicial cause, an advocate, pleader, intercessor". This is Christ's High Priestly work of interceding on our behalf to the Father. This is a gift and a privilege, yet it is only a reality for believers in Jesus Christ. Would anyone say that Jesus Christ is interceding for unbelievers? This is extremely important to understand because it lays the foundation by which we should interpret John's use of "whole world" in the next verse. 
2. The use of the word "propitiation" in verse 2 is very important and must be considered carefully lest we miss the meaning of the text. Propitiation, ἱλασμός (hilasmos), is a word to describe an appeasing of God's wrath. Jesus Christ is the propitiation for our sins because He gave himself up for us, and took our place on the cross under the wrath of God. This substitutionary sacrifice of the sinless Lamb satisfied or appeased God's wrath on behalf of the believer. To say that Jesus Christ is the propitiation for the sins of the whole world, meaning every human being in the world, is to say that the sins of every unbeliever have been paid for. If this is true, then how could you avoid universalism? For what reason is the unbeliever going to hell if Christ is the propitiation for his sins? 
Therefore, I believe that John's use of "whole world" is actually in reference to the fact that God's elect are spread throughout every tribe, nation, and tongue of the world. The gospel message and hope is not only for Jews, but for the Gentiles as well!