I am investigating the following limit $$\lim_{u\to \infty}\frac{\Gamma \left(\frac{1}{2 H},\frac{H 2^{\left(\frac{1}{2H}\right)}}{u^{\frac{1}{H}}\left(1-H\right)^{\left(2 - \frac{1}{H}\right)}} \right)}{\Gamma\left(\frac{1}{2H}\right)u^{\left(1 -H\right)\left(\frac{2}{\alpha} - 1\right)}\Gamma\left(\frac{1}{2}, \frac{1}{2}\left(H^{-2 H} (1-H)^{2 H-2} u^{2-2 H}-2 H^{-H} (1-H)^{H-1} u^{1-H}+1\right)\right) }$$
Where $0\le H < 1$ and $0\le\alpha<2$, and $\Gamma \left(z\right)$ is the usual complete Gamma function and $\Gamma \left(a, z\right) = \int_z^{\infty}t^{a-1}e^{-t}dt$.
I have noticed that as $u\to \infty$, $\Gamma \left(\frac{1}{2 H},\frac{H 2^{\left(\frac{1}{2H}\right)}}{u^{\frac{1}{H}}\left(1-H\right)^{\left(2 - \frac{1}{H}\right)}} \right)\to \Gamma\left(\frac{1}{2H}\right)$, so I think I can just focus on
$$\lim_{u\to \infty}\frac{1}{u^{\left(1 -H\right)\left(\frac{2}{\alpha} - 1\right)}\Gamma\left(\frac{1}{2}, \frac{1}{2}\left(H^{-2 H} (1-H)^{2 H-2} u^{2-2 H}-2 H^{-H} (1-H)^{H-1} u^{1-H}+1\right)\right) }$$
but under the bounds of $H$ and $\alpha$ as $u\to \infty$, $u^{\left(1 -H\right)\left(\frac{2}{\alpha} - 1\right)} \to \infty$ and $\Gamma\left(\frac{1}{2}, \frac{1}{2}\left(H^{-2 H} (1-H)^{2 H-2} u^{2-2 H}-2 H^{-H} (1-H)^{H-1} u^{1-H}+1\right)\right) \to 0$.
So if my analysis is correct this function's terms are asymptotically tending to $\frac{1}{\infty \times 0}$.
I expect this limit to be finite, but I am not sure how to show that.
Any help of advice would be much appreciated.