Regular variation

Remarks

For such $f$ there is a real number $\rho$ with $g(a) = a^\rho$ for all $a > 0$; this is Karamata's characterization theorem. The number $\rho$ is the index of $f$. Equivalently, $f(x) = x^\rho L(x)$ where $L$ is a slowly varying function. The regularly varying functions of index $0$ are exactly the slowly varying functions.

Example: $f(x) = x^2 \log(e + x)$ is regularly varying of index $2$. Indeed $f(ax)/f(x) = a^2 \cdot \log(e + ax)/\log(e + x)$, and the second factor tends to $1$.

Non-example: $f(x) = e^x$ is not regularly varying. For $a = 2$ the ratio $f(2x)/f(x) = e^x$ tends to $\infty$, so the limit is not finite.