I read in my course book that asymptotes have at least two intersection with algebraic curve at infinity. How can I take this fact in my head in a visualized way? And what does that at least means? What are these multiple intersections on infinity ?

  • $\begingroup$ Can you be more specific with what you mean by 'algebraic curve?' Because asymptotes, by definition, do not intersect their curves. $\endgroup$ Sep 18 '18 at 5:05
  • $\begingroup$ No. You can't just "assume it is intersecting at infinity". They don't intersect by definition. If you mean that the distance between the asymptote and the curve approaches zero at a point at infinity, then you need to be more precise in your question. $\endgroup$
    – YiFan
    Sep 18 '18 at 6:09

One can say that this is the definition of an asymptote; since an asymptote is a tangent to the curve at a point at infinity, the order of contact between the curve and the asymptote is higher than zero, which can be viewed as a multiple intersection. See the two references here.

  • $\begingroup$ i wanna know why does the curve and the straight line(asymptote) has atleast two intersection at infinity? See this article on [wikipedia][en.wikipedia.org/wiki/Asymptote#Algebraic_curves ]; where same thing about intersection is said. $\endgroup$
    – Vicrobot
    Sep 25 '18 at 17:30
  • $\begingroup$ Have you read the article that I linked to? $\endgroup$
    – Maxim
    Sep 25 '18 at 21:00

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