Powers Drown Logarithms

Theorem
Let $$r \in \Q^*_+$$ be a strictly positive rational number.

Then:
 * $$\lim_{x \to \infty} x^{-r} \ln x = 0$$


 * $$\lim_{y \to 0_+} y^{r} \ln y = 0$$

Proof of First Result
From Upper Bound of Natural Logarithm:

When $$x > 1$$:
 * $$\forall s \in \R: s > 0: \ln x \le \frac {x^s} s$$

Given that $$r = 0$$, we can plug $$s = \frac r 2$$ in:

$$ $$ $$

From Power of Reciprocal:
 * $$\lim_{x \to \infty} x^{-r} \frac 1 {x^{r/2}} = 0$$

and so:
 * $$\lim_{x \to \infty} x^{-r} \ln x = 0$$

by the Squeeze Theorem.

Proof of Second Result
Put $$y = \frac 1 x$$ in the first result.