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.