Combinatorics converted to a function

By Stirling's approximation, $n!\sim\sqrt{2\pi n}(\dfrac ne)^n$
$\binom{2n}n=\dfrac{(2n)!}{(n!)^2}\sim\dfrac{\sqrt{4\pi n}(\dfrac {2n}e)^{2n}}{(\sqrt{2\pi n}(\dfrac ne)^n)^2}=\dfrac 1{\sqrt\pi}\dfrac{4^n}{n^{1/2}}$
$\dfrac AB=\dfrac 4{1/2}=8$

$\displaystyle \lim_{n \to \infty} \dbinom{2n}{n} \text{ can be written as } \dfrac{4^n}{\sqrt{n\pi}} \\ \text{Comparing with } k\dfrac{A^n}{n^B} \\ A=4 \text{ and } B=\dfrac12 \\ \dfrac{A}{B} = \color{royalblue}{\boxed{8}}$