Slowly varying function

Remarks

Example: $L(x) = \log(e + x)$ is slowly varying. For fixed $a > 0$ and large $x$, $\log(e + ax) = \log x + \log a + o(1)$ and $\log(e + x) = \log x + o(1)$, so the ratio tends to $1$. Every positive measurable function with a positive finite limit at infinity is slowly varying.

Non-example: $L(x) = x^\rho$ with $\rho \ne 0$ is not slowly varying, because $L(ax)/L(x) = a^\rho \ne 1$ for $a \ne 1$.

The uniform convergence theorem says that the limit is uniform for $a$ in each compact subset of $(0, \infty)$. In the language of regular variation, slowly varying functions are the regularly varying functions of index $0$.