# Understanding limits for the polylogarithmic function

Wikipedia lists 7 different limits for the polylogarithmic function which can be found here. For example, the fourth and sixth listed limits state that $$\lim_{\mathrm{Re}(z) \rightarrow \infty} \mathrm{Li}_{-n} (\mathrm{e}^z) = (-1)^{n + 1} \mathrm{e}^{-z}, \qquad n = 1,2,3,\ldots$$ and $$\lim_{\mathrm{Re}(s) \rightarrow -\infty} \mathrm{Li}_s (\mathrm{e}^z) = \Gamma (1 - s) (-z)^{s - 1}, \qquad -\pi < \mathrm{Im}(z) < \pi,$$ respectively.

If $s$ and $z$ are both real, how is one to interpret/understand either of these limits? What I mean is, after having taken the limit, the quantity the limit was taken for, re-appears. So in the first limit $z$ re-appears after having taken a limit with respect to $z$ while in the second $s$ re-appears after having taken a limit with respect to $s$.

It means $$\text{lim}_{\text{Re}(z)\rightarrow \infty} \frac{\text{Li}_{-n}(\exp(z))}{(-1)^{n+1}\exp(-z)}=1$$