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What are asymptotes?

What are asymptotes? Here are some definitions.

Noun
  1. plural of asymptote
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Examples
Version 4 of FX-Graph does add asymptotes while Autograph expects that students will add their own vertical asymptotes.
Two straight lines, the asymptotes of the curve, pass through the geometric centre.
In book two Apollonius investigates how hyperbolas are related to their asymptotes, and he also studies how to draw tangents to given conics.
These models, which more closely resemble reality, involve one or two curves tending towards asymptotes.
I mean I liked it as much as I like finding the centres, vertices, foci, and asymptotes of various hyperbolic graphs.
The distributions become approximately exponential when the curve shown here asymptotes.

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