# Trying to solve: $$\displaystyle\lim_{{{x}\to\infty}}{\frac{{{{\ln}^{{{1000}}}{x}}}}{{{x}^{{5}}}}}$$

Trying to solve:
$\underset{x\to \mathrm{\infty }}{lim}\frac{{\mathrm{ln}}^{1000}x}{{x}^{5}}$
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Terzago66cl
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
$\frac{{\mathrm{ln}}^{b}x}{{x}^{a}}={\left(\frac{\mathrm{ln}x}{{x}^{\frac{a}{b}}}\right)}^{b}={\left(\frac{b}{a}\right)}^{b}{\left(\frac{\mathrm{ln}\left({x}^{\frac{a}{b}}\right)}{{x}^{\frac{a}{b}}}\right)}^{b}$