Regular variation

Summary

A positive function $f$ on $(0, \infty)$ is regularly varying if $f(ax)/f(x)$ has a positive finite limit as $x \to \infty$ for every $a > 0$. The limit is then always a power $a^\rho$, and $f(x) = x^\rho L(x)$ with $L$ a slowly varying function. So regularly varying functions behave like powers up to a slowly varying factor. They are the natural comparison functions in Karamata's Tauberian theorem.