In (9.23) of https://arxiv.org/abs/1908.08875 , they claim that it is "easy" to show the following limits $$\lim_{y\to 0}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}=\lim_{y\to \infty}(-y)^mq^\frac{(1-r)m}{2}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}=1\,,\quad r,m\in\mathbb{Z}\,,\quad (z;q)\equiv\prod_{k=0}^\infty(1-zq^k)\,.$$
Is this true? If so, I would be grateful for hints to prove it. Numerical tests suggest that the first limit is false. I'm inclined to say that $$\lim_{y\to 0}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}=\frac{1}{\infty}=0\,.$$