Limit of Power of x by Absolute Value of Power of Logarithm of x/Corollary
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Corollary to Limit of Power of x by Absolute Value of Power of Logarithm of x
Let $k$ be a positive real number.
Let $n$ be a positive integer.
Then:
- $\ds \lim_{x \mathop \to 0^+} x^k \paren {\ln x}^n = 0$
Proof
From Limit of $x^\alpha \size {\ln x}^\beta$, we have:
- $\ds \lim_{x \mathop \to 0^+} x^k \size {\ln x}^n = 0$
For $0 < x \le 1$, we have:
- $\ln x \le 0$
so by the definition of the absolute value, we have:
- $\size {\ln x} = -\ln x$
so:
- $\ds \lim_{x \mathop \to 0^+} x^k \paren {-\ln x}^n = 0$
That is, from the Multiple Rule for Limits of Real Functions:
- $\ds \paren {-1}^n \lim_{x \mathop \to 0^+} x^k \paren {\ln x}^n = 0$
giving:
- $\ds \lim_{x \mathop \to 0^+} x^k \paren {\ln x}^n = 0$
$\blacksquare$