From the second item under the Asymptotic Behavior section of the Wikipedia article for the incomplete gamma function, it is written that
$$ \frac{\Gamma(s,x)}{x^s} \to -\frac{1}{s} \text{ as } x \to 0 \text{ and } \Re(s) < 0 $$
where the (upper) incomplete gamma function $\Gamma$ is defined as
$$ \Gamma(s,x) := \int_x^\infty t^{s-1} e^{-t} \,\mathrm{d}t $$
How do you derive this asymptotic behavior (at least when $s$ is real)? There doesn't seem to be a reference for this statement. Maybe it can be derived from the following statement, also from the Wikipedia article?
$\Gamma(s,x) \sim \Gamma(s) - \sum_{n=0}^\infty (-1)^n \frac{x^{s+n}}{n!(s+n)}$ as an asymptotic series where $x\to0^+$ and $s\neq 0,-1,-2,\dots$