Limit of Power of x by Absolute Value of Power of Logarithm of x/Corollary

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:

$\ds \lim_{x \mathop \to 0^+} x^k \paren {\ln x}^n = 0$

$\blacksquare$