Also, I recently encountered a definition for 'asymptotes' in an old engineering mathematics book that says,"An asymptote is a straight line which cuts a curve in two points at an infinite distance from the origin and yet is not itself wholly at infinity.".
I understand very well that this definition is outdated as per modern authors. Yet, I had trouble understanding what does it mean by the use of the terms 'cuts a curve in two points'.
What prompted me to ask the question in the title is, I think by using the terms 'two points' the author might have meant the plane curve (2D). Thus I want to know if the concept of asymptotes can be applied to 3D curves.
Given the function $f(x)$ and an asymptote $a(x)=mx+n$ such that
$$\lim_{x\to \pm \infty} \frac{f(x)}{mx}=1 \quad \lim_{x\to \pm \infty} f(x)-mx=n$$
in this sense we can say that they “intersect at the two points at $\infty$ distance” but of course that is not a rigorous way to express this fact.
Of course we could have also asymptotes for a curve in 3D and also in this case we can have no more than two “intersection point at $\infty$”. Indeed this depends upon the fact that the straight line and curves are one dimensional.