I have to confess I was very amused when my friend said he understood the other four points but not unconditional election. In my experience, a "four-point" Calvinist stumbles at limited atonement. There are a lot of "TU-IP"s out there.
I do think a major stumbling block to limited atonement is the name. It would have been far better to call it definite atonement, if only TUDIP made a clever acronym. Particular atonement is good, too, and may be a viable option today as Tu Pip sounds like a rapper. Oh well...
The Westminster Confession states the doctrine:

Wherefore, they who are elected, being fallen in Adam, are redeemed by Christ, are effectually called unto faith in Christ by His Spirit working in due season, are justified, adopted, sanctified, and kept by His power, through faith, unto salvation. Neither are any other redeemed by Christ, effectually called, justified, adopted, sanctified, and saved, but the elect only.

It logically follows on the heels of unconditional election - if God chose before the foundation of the world whom He would save, why would He send His Son to die for those He never planned to save? How could He come to redeem those He did not come to redeem? Eh?

This was my stumbling block. I really struggled with all the "world" and "all" passages. How could the atonement be limited in scope when He is "the Lamb of God who takes away the sin of the world"? (John 1:29) Didn't "God so love the world that He gave His only begotten Son... that the world through Him might be saved"? (John 3:16-17)

Two things broke me out of this. First, I knew I was arguing against the limited scope of the atonement. But what was I arguing for? If He died for all (2 Corinthians 5:14-15), bringing salvation for all people (Titus 2:11), then everyone is saved. The choice was either limited atonement or universalism. Second, and this is where I want to park today, I did not understand the infinite value of Christ's atonement.

But you denied the Holy One and the Just, and asked for a murderer to be granted to you, and killed the Prince of life, whom God raised from the dead, of which we are witnesses.

Acts 3:14, 15

But we speak the wisdom of God in a mystery, the hidden wisdom which God ordained before the ages for our glory, which none of the rulers of this age knew; for had they known, they would not have crucified the Lord of glory.

1 Corinthians 2:7, 8

Therefore take heed to yourselves and to all the flock, among which the Holy Spirit has made you overseers, to shepherd the church of the Lord and God which He purchased with His own blood.

Acts 20:28

Limited atonement teaches a limit only in scope - a numerical limit, if you will. Not a limit in value. This is the blood of the Prince of Life, the Lord of Glory. This is the very blood of God. The blood of God is absolutely sufficient to have saved every person, had that been its purpose.

To make atonement for a sin against an infinite God required an infinite sacrifice.

This is not an issue of sufficiency, but of efficiency. In a sense, it is the doctrine of universal atonement (the opposing, Arminian view) that should have been labeled limited atonement. If an atonement can be applied to all men, yet some are lost, then its a very limited atonement, in my opinion.

Let me illustrate: Imagine that I had a large debt that I could not pay. I then learned that the creditors would be coming to collect. I would certainly be afraid. What if my friend - a very good friend - told me that he had already paid my debt. I should rejoice. But what if I did not rejoice. What if I did not believe him? What if I chose instead to live in my fear, wondering how I could avoid the creditors?

But what would I be fearing? Would there be a ground for my fear? If my friend had indeed paid my debt, would the creditors still come? They've been paid. They have no reason to come after me. They have their money. I would be fearing a myth.

So, if atonement has been made universally, everyone's debt has been paid. No one is going to hell. That would be double jeopardy. Is God that unjust?

As B. B. Warfield said, "The things we have to choose between are an atonement of high value, or an atonement of wide extension. The two cannot go together."

It is the Arminian who limits the atonement and dares rob the power from the blood.

More later...