I've seen several papers on heavy-tail analysis use "leading term" to refer to something which is a factor, not a term. For instance, if we've determined the asymptotics of a tail function to be \begin{equation} P(X>x)\sim\frac{1}{\log x}\cdot x^{-\log x} \quad\text{as }x\to\infty, \end{equation} they'll say that the "leading term" is $x^{-\log x}$. I get that $x^{-\log x}$ decays faster than $\frac{1}{\log x}$, so in a way it "contributes" more to the leading order behavior. Indeed, in the image below, we see that the graph of $\frac{1}{\log x}\cdot x^{-\log x}$ is much closer to $x^{-\log x}$ than to $\frac{1}{\log x}$. But what is the most sensible mathematical way of stating this? I.e. the fact that for large $x$, it's the leading factor that determines the asymptotic behavior, and the slower-decaying factor is "negligible" in some sense?
