$${n}!\approx\left(\frac{{n}}{\mathrm{e}}\right)^{{n}} \sqrt{\mathrm{2}\pi{n}} \\$$$$\left(\frac{\left(\mathrm{4}{n}\right)!}{\left(\mathrm{3}{n}\right)!}\right)^{\mathrm{1}/{n}} \approx\left(\frac{\left(\frac{\mathrm{4}{n}}{\mathrm{e}}\right)^{\mathrm{4}{n}} \sqrt{\mathrm{8}\pi{n}}}{\left(\frac{\mathrm{3}{n}}{\mathrm{e}}\right)^{\mathrm{3}{n}} \sqrt{\mathrm{6}\pi{n}}}\right)^{\mathrm{1}/{n}} =\frac{\left(\frac{\mathrm{4}{n}}{\mathrm{e}}\right)^{\mathrm{4}} }{\left(\frac{\mathrm{3}{n}}{\mathrm{e}}\right)^{\mathrm{3}} }\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{1}/{n}} = \\$$$$=\frac{\mathrm{256}{n}}{\mathrm{27e}}\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{1}/{n}} \\$$$$\Rightarrow\:\mathrm{we}\:\mathrm{have}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\:\left(\mathrm{ln}\:\left(\frac{\mathrm{256}}{\mathrm{27e}}\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{1}/{n}} \right)\right)= \\$$$$=\mathrm{ln}\:\frac{\mathrm{256}}{\mathrm{27e}}\:=\mathrm{8ln}\:\mathrm{2}\:−\mathrm{3ln}\:\mathrm{3}\:−\mathrm{1} \\$$
$${Stirling}'{s}\:{approximation} \\$$$$\left({n}\right)!\sim\:{n}^{{n}} {e}^{−{n}} =\left(\frac{{n}}{{e}}\right)^{{n}} \\$$$${Proof}: \\$$$${ln}\left({x}!\right)=\underset{{y}=\mathrm{1}} {\overset{{x}} {\sum}}{ln}\left({y}\right)=\underset{{y}=\mathrm{1}} {\overset{{x}} {\sum}}{ln}\left({y}\right)\Delta{y}\approx\underset{\mathrm{1}} {\overset{{x}} {\int}}{ln}\left({y}\right){dy} \\$$$$\Delta{y}=\mathrm{1} \\$$$$\underset{\mathrm{1}} {\overset{{x}} {\int}}{ln}\left({y}\right){dy}={yln}\left({y}\right)\mid_{\mathrm{1}} ^{{x}} −\underset{\mathrm{1}} {\overset{{x}} {\int}}{yd}\left({ln}\left({y}\right)\right)= \\$$$$={xln}\left({x}\right)−\underset{\mathrm{1}} {\overset{{x}} {\int}}{dy}={xln}\left({x}\right)−{x}+\mathrm{1} \\$$$${x}\gg\mathrm{1},\:{ln}\left({x}!\right)\sim{ln}\left({x}^{{x}} {e}^{−{x}} \right) \\$$$${Q}.{E}.{D} \\$$$$\\$$$$\underset{{n}\rightarrow\infty} {{lim}}\left[{ln}\left(\frac{\sqrt[{{n}}]{\frac{\left(\mathrm{4}{n}\right)!}{\left(\mathrm{3}{n}\right)!}}}{{n}}\right)\right]= \\$$$$=\underset{{n}\rightarrow\infty} {{lim}}\left[{ln}\left(\frac{\sqrt[{{n}}]{\left(\frac{\mathrm{4}^{\mathrm{4}} {n}^{\mathrm{4}} {e}^{\mathrm{3}} }{\mathrm{3}^{\mathrm{3}} {n}^{\mathrm{3}} {e}^{\mathrm{4}} }\right)^{{n}} }}{{n}}\right)\right]= \\$$$$=\underset{{n}\rightarrow\infty} {{lim}}\left[{ln}\left(\frac{\mathrm{256}{n}}{\mathrm{27}{ne}}\right)\right]={ln}\left(\mathrm{256}/\mathrm{27}\right)+{ln}\left(\mathrm{1}/{e}\right)= \\$$$$={ln}\left(\frac{\mathrm{256}}{\mathrm{27}}\right)−\mathrm{1} \\$$$$\\$$
$${log}\left(\frac{\sqrt[{{n}}]{\frac{\left(\mathrm{4}{n}\right)^{\mathrm{4}{n}} {e}^{−\mathrm{4}{n}} \sqrt{\mathrm{8}\pi{n}}}{\left(\mathrm{3}{n}\right)^{\mathrm{3}{n}} {e}^{−\mathrm{3}{n}} \sqrt{\mathrm{6}\pi{n}}}}}{{n}}\right)={log}\left(\frac{\mathrm{256}}{\mathrm{27}{e}}\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{{n}}} \right) \\$$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{log}\left(\frac{\mathrm{256}}{\mathrm{27}{e}}\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\frac{\mathrm{1}}{{n}}} \right)={log}\left(\frac{\mathrm{256}}{\mathrm{27}{e}}\right) \\$$