asymptotic equivalence

Sometimes two complicated quantities, while never exactly equal, march in lockstep as some variable grows large — their ratio settles to 1. Asymptotic equivalence is the precise way to say 'these two behave the same in the limit'. We write f(x) ~ g(x), read 'f is asymptotic to g', and it means that for large x (or near whatever limit is intended) the simpler g captures the true leading behaviour of f.

The precise meaning: f(x) ~ g(x) as x tends to a limit means the ratio f(x)/g(x) tends to 1. Equivalently, the difference f - g is little-o of g, that is, the error is negligible compared with g itself. So the harmonic-number sum 1 + 1/2 + ... + 1/n ~ ln n, because the ratio of the sum to ln n approaches 1 even though their difference (which approaches the Euler-Mascheroni constant, about 0.5772) never goes away. The relation is symmetric and transitive, so it behaves like a genuine notion of 'eventually the same size and shape'.

This is the headline result of most asymptotic analysis: the prime-counting function pi(x) ~ x / ln x (the prime number theorem), and Stirling's formula n! ~ sqrt(2 pi n) (n/e)^n. The crucial caveat is that ~ controls only the relative error, not the absolute error. n! ~ sqrt(2 pi n)(n/e)^n is superb as a ratio, yet the absolute gap between the two sides is astronomically large for big n — both sides are simply enormous. Equivalence promises a good percentage match, never a small numerical difference.