I was interested in the limit definition of $e$, where as $x$ tends towards infinity, the limit of $1$ plus $x$ to the negative first power, all to the power of $x$, is in fact $e$. I was wondering if the limit of the ratio between the factorial function and Stirling's approximation would converge in a similar manner. The formula in question and the Wolfram Alpha result are: $$ \lim_{x\to\infty} (\frac {x!}{\sqrt{2\pi x}(\frac{x}{e})^x})^x = \sqrt[12]{e} $$
Wolfram Alpha evaluated this limit as being the 12th root of $e$. My question being, why is this? I understand that this would normally be equal to $e$ itself, but what is it about the difference between this approximation and the actual factorial function that takes $e$ to the 12th root? Why 12 of all roots? Why not 13? or 14? I'm sure there is an explanation for this, I am just unsure why.