Is there a formal definition for $f(x)$ ~ g(x)? I was looking to see if curved asymptotes were possible and came across an answer that referred to an end behavior of a function as being $f(x)$ ~ $x^2$.  I'm assuming this either means the end behavior of a function or a generalization of what a function does given a set of large x values, but I don't want to simply assume something I don't know.  Is there a formal definition for what $f(x)$ ~ $g(x)$ is?

$f(x) \sim g(x)$, I believe, is formally defined as $$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1$$
Take $f(x) = x^2$ and $g(x) = x^2 - x$ for example. $f(x) \sim g(x)$ because $$\lim_{x \to \infty} \frac{x^2}{x^2-x} = \lim_{x \to \infty} \frac{x}{x-1} = 1$$
