For the sake of this post, we only look at real-valued gamma functions.
We know that $\lim_{b \to \infty}\gamma(a,b)=\Gamma(a)$, so we have
$$\lim_{b \to \infty} \frac{\gamma(a,b)}{\Gamma(a)}=1.$$
How to prove asymptotic limit of an incomplete Gamma function shows that
$$\lim_{a \to \infty} \frac{\gamma(a,b)}{\Gamma(a)}=0.$$
I also remembered one result, although I cannot recall the reference:
$$\lim_{a \to \infty} \frac{\Gamma(a,a)}{\Gamma(a)}=\frac{1}{2}.$$
Therefore, we also have $$\lim_{a \to \infty} \frac{\gamma(a,a)}{\Gamma(a)}=\frac{1}{2}.$$
I am curious about the following limit, can we conjecture that for general $a>0,b>0$: $$\lim_{k \to \infty} \frac{\gamma(ak,bk)}{\Gamma(ak)}=\frac{1}{2}?$$
Thank you.