Asymptotic equivalence

Functions $f$ and $g$ , with $g\ne0$ near the limiting point, are asymptotically equivalent, written $f\sim g$ , if the following limit holds:

\lim \frac{f}{g}=1.

For real or complex functions this is equivalent to $f-g=o(|g|)$ , using little-o notation. The limiting variable and direction are part of the assertion.

Distinguishing two conventions

Some authors use $\sim$ for a two-sided bound. Here it always means ratio tending to one; comparability is written $\asymp$ . For example, $2\varepsilon\asymp\varepsilon$ but $2\varepsilon\not\sim\varepsilon$ .

References

NIST Digital Library of Mathematical Functions, §2.1: Definitions and elementary properties.