Powers Drown Logarithms
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Theorem
Let $r \in \R_{>0}$ be a (strictly) positive real number.
Then:
- $\ds \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$
Corollary
- $\ds \lim_{y \mathop \to 0_+} y^r \ln y = 0$
Proof
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:
\(\ds x^{-r} \ln x\) | \(=\) | \(\ds x^{-r/2} \paren {x^{-s} \ln x}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \frac {x^{-r/2} } s\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds s \frac 1 {x^{r/2} }\) |
From Sequence of Powers of Reciprocals is Null Sequence:
- $\ds \lim_{x \mathop \to \infty} x^{-r} \frac 1 {x^{r/2} } = 0$
and so:
- $\ds \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$
by the Squeeze Theorem for Real Sequences.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.3 \ (2) \ \text{(i)}$