Let $S \subseteq \mathbb{R}^n$ be an unbounded semialgebraic set. Is there a standard accepted definition of what it means for a linear-affine subspace $L$ of $\mathbb{R}^n$ to be asymptotic to $S$? The (vague) general idea is that points of $L$ grow arbitrarily to $S$, but this seems unclear. It seems too restrictive to require that the points of $L$ should be within any prescribed distance from $S$ outside of some bounded set... But then do we need a notion of "branch at infinity" for semialgebraic sets?
Moreover, given some reasonable definition of asymptote, how in general does one find all asymptotes of $S$?