My textbook states the following theorem:

If $r>0$ is a rational number, then $$\lim_{x \to \infty}\frac{1}{x^r}=0$$ If $r>0$ is a rational number such that $x^r$ is defined for all $x$, then $$\lim_{x \to -\infty}\frac{1}{x^r}=0$$

I'm comfortable with these statements (and can prove them using the epsilon-delta definition for limits at infinity), but I'm wondering if this theorem can be extended to irrational $r$. I'm confident that it can as $x \to \infty$; however, I'm not sure how to conceptualize a negative number raised to an irrational power, let alone determine if $x^r$ is defined for all $x$ if $r$ is irrational. Does anyone know how to approach this? Is this question beyond the scope of calculus?