Question Number 188798 by mathlove last updated on 07/Mar/23
Answered by ARUNG_Brandon_MBU last updated on 07/Mar/23
![L=lim_(x→0) ((2^x 3^x 4^x 5^x ...n^x −1)/x) =lim_(x→0) ((A^x −1)/x) ,[ A=n! ] =lnA=ln(n!)](https://www.tinkutara.com/question/Q188802.png)
$$\mathscr{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}^{{x}} \mathrm{3}^{{x}} \mathrm{4}^{{x}} \mathrm{5}^{{x}} …{n}^{{x}} −\mathrm{1}}{{x}} \\ $$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{A}^{{x}} −\mathrm{1}}{{x}}\:,\left[\:{A}={n}!\:\right] \\ $$$$\:\:\:\:\:=\mathrm{ln}{A}=\mathrm{ln}\left({n}!\right) \\ $$