Asymptote Definition and 8 Discussions

In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.There are three kinds of asymptotes: horizontal, vertical and oblique. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.
More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.
Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.

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  1. S

    Question about asymptotes of rational function

    I tried graphing the function in the calculator, and the graph seems to have a horizontal asymptote at y=0, not at y=1. Why is this so? Thanks for helping out.
  2. G

    Different types of hyperbolas and their properties

    I know the hyperbola of the form x^2/a^2-y^2/b^2=1 and xy=c; but coming across this question I'm put in a dilemma of how to proceed with calculating anything of it - say eccentricity or latus rectum or transverse axis as said. How to generalize a hyperbola (but i don't want a complex derivation...
  3. Ranku

    I Asymptotic Hubble constant

    For constant dark energy, Hubble value will eventually become asymptotic. If dark energy were dynamic and gently decreasing, what will the value of Hubble eventually become - will it asymptote or keep decreasing?
  4. M

    Getting the Horizontal Asymptotes

    Homework Statement Homework Equations The Attempt at a Solution I understand there is no vertical asymptotes and can usually get the horizontal ,but can't understand with the exponential.
  5. K

    Draw the graph of arctan((x-1)/(x+1))

    I am so far able to find the domain, intercepts, concavity and increase/decrease. But I am stuck at finding the asymptotes for the graph. I think there is no vertical asymptote or is there one at x=-1? I think the horizontal asymptote is y=pi/4
  6. J

    How to know if this irrational function has no asymptotes?

    1. The problem statement, all variables and given/known dat F(x)=x+1-3sqrt((x-1)/(ax+1)) For which value of a ,(c) has no asymptote? Homework Equations The Attempt at a Solution I know if a>0 then (c) will have 2 asymptote And if a<o then (c) will have 1 vertical asymptote. But I can't...
  7. T

    Asymptotes of (X-5)/(X^2+X-6)

    Good afternoon 1. Homework Statement Draw graph of the following equation Homework Equations \frac{X-5}{X^2+X-6} = y The Attempt at a Solution my problem is searching for the vertical asymptote from what I know the way to find the vertical asymptote is by limiting the equation near to ∞...
  8. D

    Can time be modeled by an asymptote?

    Hello all! I'm new to this forum (and forums in general as this is my first) so please excuse my etiquette. When I was in my trig class (I'm a high schooler), we brushed up upon asymptotes, and it made me wonder: Can time be modeled by an asymptote? I like the idea of a line moving in a...