Powers Drown Logarithms

Theorem
Let $r \in \Q_{>0}$ be a strictly positive rational number.

Then:
 * $(1): \quad \displaystyle \lim_{x \to \infty} x^{-r} \ln x = 0$


 * $(2): \quad \displaystyle \lim_{y \to 0_+} y^r \ln y = 0$

Proof of First Result
From Upper Bound of Natural Logarithm:

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

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

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

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

by the Squeeze Theorem.

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