Actually, to prove that $$\lim_{n\to\infty}\frac{n^2}{n+\frac{1}{2}}\left[\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2=1,$$ there is no need in the Bohr's correspondence principle. Stirling's Series can be used to get $$\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)}\sim z^{\alpha-\beta},$$ as $z\to\infty$ (see, for example, https://projecteuclid.org/euclid.pjm/1102613160 - The asymptotic expansion of a ratio of gamma functions, by A. Erdelyi and F. G. Tricomi). Then $$\lim_{n\to\infty}\frac{n^2}{n+\frac{1}{2}}\left[\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2=\lim_{n\to\infty}\frac{n^2}{(n+\frac{1}{2})n}=1.$$

P.S. This answer was inspired by Paul Garrett's comment about Watson's Lemma in this MO question: The sum of an hydrogen atom related infinite series

Actually, to prove that $$\lim_{n\to\infty}\frac{n^2}{n+\frac{1}{2}}\left[\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2=1,$$ there is no need in the Bohr's correspondence principle. Stirling's Series can be used to get $$\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)}\sim z^{\alpha-\beta},$$ as $z\to\infty$ (see, for example, https://projecteuclid.org/euclid.pjm/1102613160 - The asymptotic expansion of a ratio of gamma functions, by A. Erdelyi and F. G. Tricomi). Then $$\lim_{n\to\infty}\frac{n^2}{n+\frac{1}{2}}\left[\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2=\lim_{n\to\infty}\frac{n^2}{(n+\frac{1}{2})n}=1.$$

P.S. This answer was inspired by Paul Garrett's comment about Watson's Lemma in this MO question: The sum of an hydrogen atom related infinite series