How to show that $$S =\lim_{n \to \infty} \frac{\Gamma(\frac{n+1}2)}{\Gamma(\frac n2)\sqrt{n}}$$ exists? If it exists I can show that is equal to $\displaystyle \frac 1{\sqrt 2}$ [0], but I don't really know how to prove that it exists.
Thank you :)
[0] Since $$\frac{\Gamma(\frac{n+1}2)}{\Gamma(\frac n2)\sqrt{n}} = \frac{\left(\frac n2 + \frac 12\right)\Gamma(\frac{n-1}2)}{\Gamma(\frac n2)\sqrt{n}} =\frac{\left(\frac n2 + \frac 12\right)\Gamma(\frac{n-1}2)\sqrt{n-1}}{\Gamma(\frac n2)\sqrt{n(n-1)}} $$
Taking limit yields $S = \frac 1{2S} \implies S = \frac 1{\sqrt 2}$
Finding a limit usually means that it exists. The trick to actually get a proof is to present your results slightly differently :
The first line of your computation does not involve limits, so you are allowed to write it without having proved that a limit exists. Now you should perform a few simplifications (which you have presumably done when taking the limits on the following line), then state that all the remaining factors have a limit and invoke the general theorems for limits to state that your whole expression has a limit. Which you have exhibited, if your computations are correct.