Trying to solve:

$\underset{x\to \mathrm{\infty}}{lim}\frac{{\mathrm{ln}}^{1000}x}{{x}^{5}}$

Jakobe Norton
2022-04-10
Answered

Trying to solve:

$\underset{x\to \mathrm{\infty}}{lim}\frac{{\mathrm{ln}}^{1000}x}{{x}^{5}}$

You can still ask an expert for help

Terzago66cl

Answered 2022-04-11
Author has **8** answers

Be careful using limit operation.

First, let show that$\underset{x\to +\mathrm{\infty}}{lim}\frac{\mathrm{ln}x}{x}=0$ . For $t\ge 1$ , we have $t\ge \sqrt{t}$ which imply for $x\ge 1$

$0\le \mathrm{ln}x={\int}_{1}^{x}\frac{dt}{t}\le {\int}_{1}^{x}\frac{dt}{\sqrt{t}}=2\sqrt{x}-2\le 2\sqrt{x}$

Then, for any$a,b>0$ and $x>1$ , we have

$\frac{{\mathrm{ln}}^{b}x}{{x}^{a}}={\left(\frac{\mathrm{ln}x}{{x}^{\frac{a}{b}}}\right)}^{b}={\left(\frac{b}{a}\right)}^{b}{\left(\frac{\mathrm{ln}\left({x}^{\frac{a}{b}}\right)}{{x}^{\frac{a}{b}}}\right)}^{b}$

which answer your question.

First, let show that

Then, for any

which answer your question.

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