The exact question I have is $$\lim\_{n\to\infty} \frac{G(n+\frac{5}{4})G(n+\frac{7}{4})^{2}G(n+\frac{9}{4})}{G(n+2)^{2}G(n+\frac{3}{2})^{2}}$$

Intuitively it looks like it should tend to $1$ but I do not have experience with the Barnes G-function.