We know that $$e_q(z)=\sum_{j\geq 0} \frac{z^j}{(q;q)_j}=\frac{1}{(z;q)_{\infty}}$$
where $(a;q)_{\infty}=\prod_{i=0}^{\infty}(1−aq^i)$ denotes the q-shifted factorial.
The limit between the $q$-exponential and the ordinary exponential is
$$\lim_{q\rightarrow1}e_q((1-q)z)=\lim_{q\rightarrow1}\frac{1}{((1-q)z;q)_{\infty}}=e^z.$$
My question is how to prove that
$$\lim_{q\rightarrow1}\frac{1}{(a\sqrt{q(1-q)};q)_{\infty}}=e^{\tfrac{-1}{2}a^2}?$$
I'm glad for your help.