Christ died for sinners

In 5-point Calvinism, is limited atonement and/or limited election in tension with the universal offer of the gospel? 

i) God doesn't directly offer the gospel to every individual, or directly command every individual to believe the gospel. 

In that respect, the offer of the gospel parallels special revelation. In might be more efficient if God privately revealed himself to every individual, but instead, God resorts to a public revelation. A mass medium. 

One reason, perhaps, is that humans are social creatures, so having Scripture as a common reference point is a unifying principle.

Be that as it may, the offer of the gospel is like a recipe. If you follow the instructions, this will be the result. A recipe doesn't order anyone in particular to use that recipe. 

ii) In nature, there's a principle of redundancy. For instance, a maple tree produces far more seeds (or maple copters) than will every take root and become trees in their own right. But the redundancy is purposeful. If enough maple trees produce enough airborne seeds, that greatly raises the odds that some of them will take root and produce trees in their own right.

Likewise, many animals produce multiple offspring, only a few of which survive to maturity. But in order to at least achieve a replacement rate, it's necessary to produce offspring in excess of the replacement rate, to offset the loss of the offspring that are eaten by predators before they reach sexual maturity and repeat the reproductive cycle. By the same token, multiple sperm raise the odds that one will fertilize the ovum. 

Humans imitate this principle. For instance, absent vaccination, some people will contract a serious communicable disease and some won't. Since we don't know which is which, we resort to mass vaccination to ensure, as best we can, that everyone who would be susceptible is covered. We vaccinate everyone, not because everyone needs it, but to make reasonably certain that we get the ones who do need it. It isn't necessary for everyone, but it's necessary to include more people in order to cover the subset that really need it. 

Likewise, the military might resort to more extensive bombing strikes to raise the odds of hitting the targets. Or resort to bombs with higher yield to achieve the same end. It gives you a margin of error. 

By analogy, the universal offer of the gospel will be heard by elect and reprobate alike. That's the nature of a mass medium of communication. That doesn't mean it's intended for all. Rather, that's a way of reaching the intended subset. Given that humans are social creatures, unless God privately discloses the gospel to the elect, the only alternative is a general message. 

iii) Let's consider a more subtle illustration. Suppose one country invades another country. Some of the natives form an underground resistance movement. They are planning a counterattack to oust the occupation force. But it will take a while for them to get all their ducks in a row. 

When they are ready to launch the counterattack, they have sympathizers in the news media do a public service announcement. This will seem to be a perfectly innocuous message. But will contain some code phrases that members of the resistance movement will recognize. That will be the signal to come out of hiding and strike back.

The enemy will hear the same announcement, but it won't detect the coded message embedded in the announcement. The enemy isn't privy to the code phrases. 

The message has to be broadcast nationwide to reach all the far-flung resistance cells. Everyone will hear the same message, but everyone won't register the ulterior significance of the message. 

iv) Perhaps a 4-point Calvinist would say this is parallel to the relationship between unlimited atonement and limited election. Christ dies for everyone to cover the elect. 

Whether you think that makes sense depends on your view of what the atonement targets. Does it cover sin? Sins? Or sinners? Does the death of Christ make atonement for some abstraction we call sin? Does it make atonement for sins, as distinct from the agents who committed them? Or does it make atonement for elect sinners? For their guilt?

I don't deny that Scripture sometimes speaks of making atonement for "sin" or "sins", but I think that's shorthand for sinners. I doubt Scripture intends to treat sin as an aggregate substance in abstraction from the particular agents who commit particular sins. Sin is personal. 

If Christ died for elect sinners, then it isn't necessary for the scope of the atonement to exceed the elect in order to cover the elect. If, moreover, Christ dies for the damned, then the atonement doesn't entail the salvation of anyone in particular. That greatly weakens the link between atonement and salvation.