Operations on asymptotic equivalents

The asymptotic equivalence is denoted by $\sim$ and we say that $f \sim g$ if $\lim_{x \to \infty} f(x)/g(x) = 1$

I have read that powers in $\mathbb{N}^*$ preserve the asymptotic equivalences. Do real powers preserve equivalence?

1) Let us suppose that we have $f \sim g^{c}$ where $c$ is a real positive constant. Can we write $g \sim f^{1/c}$ ?

2) What about the special case where $c=1/b$ with $b$ a positive integer?

Thank you.

• Use the definition you have. If $\frac{f}{g^c} \to 1$, then $\left(\frac{f}{g^c}\right)^{1/c} \to ?$ – Clement C. Jun 28 '17 at 14:02

1 Answer

Yes. The function $h(x)=x^{1/c}$ is continuous. Hence $$\lim_x \frac{f(x)}{g(x)}=1 \implies \lim_x h\left(\frac{f(x)}{g(x)}\right)=h\left(\lim_x \frac{f(x)}{g(x)}\right)=1.$$

• Thank you for your answer. Yes, The definitition can be applied, my mistake.... I do not understand why the property I have read refer to integer powers only. – Dingo13 Jun 28 '17 at 14:08
• I don't know your reference, but it works for every real $c>0$. – Paolo Leonetti Jun 28 '17 at 14:09