Rank functions of n by their order of growth

Question

I'd like to rank some functions according to their order of growth (e.g. n^2 < n^3). I'm not sure how to do order of growth in mathematica, so I was trying to rank the functions by their Limit[x,x->Infinity]. Here's what I have so far:

v = {2^(Log[2, n]), n^(1/2), Log[10, n], 2^(2^n), Log[2, Log[2, n]],
  n*Log[2, n], n!, Iteratedlog[2, n], 1, n^3,
  Log[2, n], (n^2)*Log[2, n], n, n^n, 100, n^2, 2^(2^(n + 1)), 2^n}

Sort[v, Limit[#1, n -> Infinity] < Limit[#2, n -> Infinity] &]

However, this doesn't even seem to work to sort the functions as a limit. What steps should I take to solve my problem?

Thanks!

Answer 1

Don't compare values that are likely to be infinities. Instead, compute Limit on ratios of arguments to be sorted:

Sort[v, Limit[#1 / #2, n -> Infinity] < 1 &]

You need to use valid functions, though; whatever Iteratedlog is, it's going to be nonsensical on this comparison as it's not defined. Mathematica silently skips complaining about it, though.

Of course, if you consider growth towards negative infinity "growth", you probably want to add Abs in the comparison function.

EDIT: Indeed treatise linked by Sektor in comments is much more rigorous. Mine just performs the comparison it directly implies: whether ratio of equations is below or above (or equal) to 1. OTOH, my code doesn't really show which functions belong to different growth classes, so you could naively say it does the job...