In interested in how fast $\Gamma(\alpha,x) \to \Gamma(\alpha)$ as $x \to 0$, for some fixed $\alpha>1$. For example, set $\beta>0$ and consider the limit
$$\lim_{x\to0}x^{-\beta}(\Gamma(\alpha,x) - \Gamma(\alpha)).$$
By L'Hopital's rule, the limit is convergent whenever $\alpha\geq \beta$.
Thus, $\Gamma(\alpha,x) \to \Gamma(\alpha)$ as $x \to 0$ as fast as $x^{-\alpha}$? Are the other conditions out there?
Thanks,
Around $x=0$, the series expansion is given by $$\Gamma (a,x)=\Gamma (a)+x^a \left(-\frac{1}{a}+\frac{x}{a+1}-\frac{x^2}{2 (a+2)}+\frac{x^3}{6 (a+3)}+O\left(x^4\right)\right)$$ or, if you prefer the infinite summation, $$\Gamma (a,x)=\Gamma (a)+x^a \sum_{k=0}^\infty \frac{(-1)^{k+1}}{k! \,(a+k)}x^k$$ I do not see any condition to be fulfilled.