Asymptotic equivalence

Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$. The functions are asymptotically equivalent as $x \to \infty$ if $f(x) - g(x) = o(g(x))$ as $x \to \infty$, in the sense of little o notation. The notation is $f(x) \sim g(x)$.

Remarks

If $g(x) \ne 0$ for large $x$, then $f \sim g$ means that $f(x)/g(x) \to 1$. The relation $\sim$ is an equivalence relation. For example, $x^2 + x \sim x^2$. Stirling's formula states $n! \sim \sqrt{2 \pi n}\, (n/e)^n$. Mathlib defines it for filters, as Asymptotics.IsEquivalent.

Explanations

L1 human

Two functions are asymptotically equivalent if their ratio tends to $1$. Their difference may still grow, but it is small compared with the functions.