TITLE: Factorial limit from gamma function calculation
QUESTION [8 upvotes]: I want to show that $$\lim_{n\rightarrow\infty}\dfrac{\Gamma\left(n+\frac12\right)}{\sqrt{n}\Gamma(n)}=1$$
Using the formula for $\Gamma\left(n+\frac12\right)$ here, it reduces to $$\lim_{n\rightarrow\infty}\dfrac{(2n)!\sqrt{\pi}}{2^{2n}\sqrt{n}(n-1)!n!}=1$$
How to prove that?

REPLY [3 votes]: See "Inequalities for Gamma Function Ratios", 
G.J.O. Jameson, American Mathematical Monthly, 
December 2013, pp 936-940.
This recent article gives elementary proofs of 
Gautschi-type inequalities. In particular, you 
get 
$$\left({x\over x+1/2}\right)^{1/2}\leq {\Gamma(x+1/2)\over\sqrt{x}\,\Gamma(x)}\leq 1.$$
Now let $x\to\infty$.