I agree with this to some extent. Another perspective: think about the element [(1, 2, 3, 4, ...)] in the ultrafilter; let's call this omega. On some level, all

I agree with this to some extent. Another perspective: think about the element [(1, 2, 3, 4, ...)] in the ultrafilter; let's call this omega. On some level, all of these questions are really just questions about what properties omega has: is it even or odd, prime or composite, etc. Simultaneously deciding all of these questions in a coherent way is equivalent to specifying an ultrafilter. Similarly, when we ask about some function f(x) being > g(x) asymptotically, we are basically asking if f(omega) > g(omega). This is just a different view of the same thing. For instance, your question happens to be equivalent to asking whether sin(omega) > cos(omega), and thus if tan(omega) > 1. This is true iff the fractional part the hyperreal number omega/(2*pi) is between 1/8 and 5/8. Thus we have reduced the asymptotic statement to a question about an arithmetical property of one particular hyperreal number. Choosing an ultrafilter basically involves simultaneously determining all properties of omega. There are different ultrafilters, each providing a different coherent "universe" which decides all possible predicates in a coherent way. That this is possible (with the axiom of choice) is highly interesting. However, it doesn't seem necessary for asymptotic analysis. Of course, if there is some "canonical" or "most natural" ultrafilter to choose from, with some magical property universally deemed important, then it would settle your question and all such questions in a natural way.