How can 1 Corinthians 15:3 be reconciled with Calvinism and limited atonement? I have been studying the 5 Points of Calvinism (TULIP) for some time and I can say that I accept 4 out of 5. 

My biggest problem with Calvinism is the doctrine of Limited Atonement that says Jesus died only for the elect. I find this view highly opposed to the Scriptures. I tried pretty hard to reconcile this doctrine with Scripture but it was too hard, there are too many places where it says that Jesus died for the whole world/all/everyone and those can be explained. What really made me lose faith in this doctrine is Paul telling to the Corinthians what the GOSPEL is:


  1 Corinthians 15:1–4 (NIV)
  Now, brothers and sisters, I want to remind you of the gospel I preached to you, which you received and on which you have taken your stand. 2 By this gospel you are saved, if you hold firmly to the word I preached to you. Otherwise, you have believed in vain.  3 For what I received I passed on to you as of first importance: that Christ died for our sins according to the Scriptures, 4 that he was buried, that he was raised on the third day according to the Scriptures,


It is clear that the gospel contains "Christ died for our sins". "OUR" in this context can refer to the elect but this preaching of the gospel was made before they accepted it so Paul told them (before they accepted it) that Jesus died for their sins.

So when we preach the gospel it is clear that we should say "Jesus died for your sins".

Some will say that Calvinists that believe the doctrine of Limited Atonement believe in a false gospel. I won't dare to say this for I know there are a lot of godly people who are Reformed (James White, Jeff Durbin, Paul Washer, etc)

Is there any way to reconcile 1 Corinthians 15:3 with Reformed Theology?

Thank you very much for taking the time to read this. :)
I've looked in numerous systematic theologies and books on evangelism written by Calvinists, and none of them address this particular issue, even though they address many other challenging verses related to the doctrine of limited atonement.

There are probably two reasons for this:


Some Calvinists are okay with saying "Christ died for your sins" to people who are not currently saved.
Paul's summary in this passage doesn't logically require that he originally preached, without conditions, "Christ died for your sins."


Regarding the first point, here's how Wayne Grudem puts it:


  I do not think we should rush to criticize an evangelist who tells an audience of unbelievers, ‘Christ died for your sins,’ if it is made clear in the context that it is necessary to trust in Christ before one can receive the 
  benefits of the gospel offer. In that sense the sentence is simply understood to mean ‘Christ died to offer you forgiveness for your sins’ or ‘Christ died to make available forgiveness for your sins.’ The important point here is that sinners realize that salvation is available for everyone and that payment of sins is available for everyone. (Systematic Theology, 601)


Of course, other Calvinists disagree with him (he spends a fair amount of space in his ST dealing with this debate).  So then how about our second point?  Is there another way to understand what Paul is saying here?

There is.  Let's look at the verse again, skipping over a few parts:


  I would remind you, brothers, of the gospel I preached to you, which you received, in which you stand [...]. 3 For I delivered to you as of first importance what I also received: that Christ died for our sins [...]. (ESV)


It's not actually logically necessary to draw the conclusion that you did.  Paul is clearly addressing believers here, people for whom Christ died.  When he preached to them, he could have said things that all Calvinists are fine with:


"Christ died for the sins of those who believe in him"
"Christ died for the sins of the elect"
"If you repent and believe, Christ's death cleanses you of your sins"


But now, because he is addressing only those who actually repented and believed, it's fine for him to say "Christ died for our sins."  This isn't inconsistent –
 it's simply referring to the current state of the current audience, and isn't a direct quote.  

Here's a (sacreligious) example.  A coach says to two sports teams, "If you win, you get a trophy."  Then the two teams play, and one wins.  Then the coach says, "Congratulations winning team.  Here's the trophy I promised you."  It's obvious that the coach hasn't lied here.  Why not?  Because he doesn't need to repeat the condition of the promise (winning) when addressing those who met that condition (the winning team).

In the same way, Paul can simply be understood here as saying, "Believers, as promised, Christ died for our sins."   Like the coach, he doesn't need to repeat the condition of the promise (repent and believe) when addressing those who met that condition (believers). 

All that to say – we can't logically exclude the possibility that Paul used conditional language (if you are elect, if you repent and believe) in his original preaching to the Corinthians.