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Question Number 34951 by rahul 19 last updated on 13/May/18

$$\underset{n\to \mathrm{\infty }}{\lim }{\left(\frac{e\times a}{n}\right)}^n=?$$$$Hereaϵ{\mathbb{R}}^{+}$$

Answered by MJS last updated on 14/May/18

$$a\mathrm{e}=p\in {\mathbb{R}}^{+}$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{p^n}{n^n}=L$$$$L=\underset{n\to \mathrm{\infty }}{\lim }\frac{p^n}{n^n}=\underset{n\to \mathrm{\infty }}{\lim }\frac{\frac{d}{dn}\left[p^n\right]}{\frac{d}{dn}\left[n^n\right]}=\underset{n\to \mathrm{\infty }}{\lim }\frac{\ln \left(p\right)\times p^n}{(1+\ln \left(n\right))\times n^n}=$$$$=L\times \underset{n\to \mathrm{\infty }}{\lim }\frac{\ln \left(p\right)}{1+\ln \left(n\right)}=L\times 0=0$$

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