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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 ?

Vicrobot
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  • Can you be more specific with what you mean by 'algebraic curve?' Because asymptotes, by definition, do not intersect their curves. – Christopher Marley Sep 18 '18 at 05:05
  • 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. – YiFan Tey Sep 18 '18 at 06:09

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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.

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