Asymptotic comparisons¶

AsymptoticLess, AsymptoticLessEqual, AsymptoticGreater, AsymptoticGreaterEqual, AsymptoticEqual, and AsymptoticEquivalent compare the growth of two expressions as a variable approaches a limit point. They are Landau notation: o, O, ω, Ω, Θ, and ~ respectively.

AsymptoticLess[f, g, x -> a] asks whether f is negligible next to g, i.e. whether f ∈ o(g):

$ wo 'AsymptoticLess[x, x^2, x -> Infinity]'
True

$ wo 'AsymptoticLess[Log[x], x, x -> Infinity]'
True

The comparison depends on where the limit is taken — near 0 the higher power is the smaller one:

$ wo 'AsymptoticLess[x, x^2, x -> 0]'
False

$ wo 'AsymptoticLess[x^2, x, x -> 0]'
True

AsymptoticLessEqual is the O relation, so unlike AsymptoticLess it holds between functions of the same order:

$ wo 'AsymptoticLessEqual[x, x, x -> Infinity]'
True

$ wo 'AsymptoticLess[x, x, x -> Infinity]'
False

It only needs the ratio to stay bounded, not to converge, so a bounded oscillation qualifies:

$ wo 'AsymptoticLessEqual[Sin[x], 1, x -> Infinity]'
True

AsymptoticGreater and AsymptoticGreaterEqual are the same relations with the arguments exchanged:

$ wo 'AsymptoticGreater[Exp[x], x^3, x -> Infinity]'
True

AsymptoticEqual is the Θ relation — bounded in both directions — so it ignores constant factors and lower-order terms:

$ wo 'AsymptoticEqual[3 x + 1, x, x -> Infinity]'
True

AsymptoticEquivalent is stricter: it requires the ratio to reach exactly 1, which a constant factor destroys:

$ wo 'AsymptoticEquivalent[x + 1, x, x -> Infinity]'
True

$ wo 'AsymptoticEquivalent[2 x, x, x -> Infinity]'
False