What is Asymptote: Definition and 80 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. P

    About critical damping resistance of a ballistic galvanometer

    At critical condition, ##\omega=0## so time period will be infinite and so will be ##\lambda##.Therefore, the critical resistance will be the corresponding resistance(plus galvanometer resistance)of the asymptote of ##\lambda## vs. ##R_2## graph(the graph is a rectangular hyperbola). But here's...
  2. chwala

    Solve the problem that involves iteration

    part (a) Asymptote at ##x=0.5## part (b) ##\dfrac{e^x}{4x^2-1}= -2## ##e^x=2-8x^2## ##2e^x= 4-16x^2## ##16x^2=4-2e^x## ##x^2= \dfrac{4-2e^x}{16}## ##x=\dfrac{\sqrt{4-2e^x}}{4}## part (c) ... ##x_{2}=0.2851## ##x_{3}=0.2894## ##x_{4}=0.2881## ##x_{5}=0.2885## ##x_{6}=0.2884##...
  3. N

    MHB Horizontal Asymptote of Rational Function

    Given f(x) = [sqrt{2x^2 - x + 10}]/(2x - 3), find the horizontal asymptote. Top degree does not = bottom degree. Top degree is not less than bottom degree. If top degree > bottom degree, the horizontal asymptote DNE. The problem for me is that 2x^2 lies within the radical. I can rewrite...
  4. greg_rack

    Horizontal asymptote of a parametric function

    I'll write my procedure: $$\lim_{x\to\infty}[\frac{(a-2)x^3+x^2}{ax^2+6x+1}]\rightarrow\frac{x(a-2)}{a}\in \mathbb{R}$$ And now, assumed that everything's correct, how do I assign ##a## a value for having that limit finite and ##\in \mathbb{R}##, and so an horizontal asymptote?
  5. karush

    MHB -s8.3.6.46 Find Slant asymptote

    $\tiny{s8.3.6.46}$ Find the Slant asymptote $y=\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$ Ok the last time I did a slant asymptote was decades ago in Algrebra but this is a calculus problem the example started with this $\displaystyle\lim_{x \to \infty}[f(x)-(mx+b)]=0$ long division returns...
  6. LCSphysicist

    Find the equation knowing its asymptote in the infinite

    Find all linear differential equations of first order that satisfy this property: All solutions are asymptotic to the straight line y = 3 - x, when x -> infinity First i began writing the general equation: y' + g(x)*y = h(x) I would say that when x-> infinity, our equations will tends to 3-x...
  7. 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.
  8. 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...
  9. M

    Control Theory - Nyquist Plot - Speed of Response from Low Frequency Asymptote

    [Moderator: moved from a homework forum. This does not sound like homework.]Homework Statement:: Why is it the case that when the low-frequency response is to the right of the M = 1 line that the 'speed of response is slow'? Relevant Equations:: M-cirlces Hi, Hope you are doing well and...
  10. S

    Asymptote of x^3 - x^5 / ( x^2 + 1) and similar curves

    Playing with some numerical simulations, I plotted this in Wolfram Cloud / Mathematica: ##x^3-\frac{x^5}{x^2+2}## I had naively expected it to approach ##x^3−x^3=0##, but that isn't the case. It approaches 2x. I can now vaguely understand that the two terms need not cancel at infinity, but I'd...
  11. 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?
  12. M

    MHB What are the steps to finding an oblique asymptote for a rational function?

    Find the oblique asymptote of f(x) = (x^2 - 16)/(x - 4). I need the steps not the solution.
  13. M

    MHB What Are the Steps to Find the Horizontal Asymptote of f(x) = (x^2 - 9)/(x - 3)?

    Find the horizontal asymptote of f(x) = (x^2 - 9)/(x - 3). I need the steps not the solution.
  14. M

    MHB Vertical Asymptote

    Find the vertical asymptote of f(x) = (5 - x^2)/(x - 3). I need the steps not the solution.
  15. M

    Getting the Horizontal Asymptotes

    Homework Statement Homework EquationsThe Attempt at a Solution I understand there is no vertical asymptotes and can usually get the horizontal ,but can't understand with the exponential.
  16. F

    Asymptote of a curve in polar coordinates

    Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...
  17. B

    MHB Need help in creating a question that uses the word asymptote

    Hello everyone. Currently I teach a test prep course that covers a multitude of subjects. One specific area of the test covers Math Algebra, Trigonometry etc. My employee is a Math teacher who doesn't understand that sometimes the students (all of whom are adults) cannot get past the...
  18. P

    Hyperbola's asymptote, where's my mistake?

    I have tried like 5 times to do this problem ans still don't get the answer I'm supposed to (A). Anybody finds mistake in my work below?
  19. 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
  20. 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 EquationsThe 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 find...
  21. 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 ∞...
  22. L

    Physical Asymptote Homework: Trajectory y=x^4-x^2 & Limit y(x)=h

    Homework Statement For example particle performs a motion in x-y plane. In y there are walls from both side so particle can go in y direction from zero to h. I need to plot trajectory. If I got trajectory y=x^4-x^2 then \lim_{x\to \infty}y(x)=\infty[/B]Homework EquationsThe Attempt at a...
  23. L

    Right asymptote of a simple function doesn't exist?

    Hi everyone, I was working on a problem recently, something related to oral absorption of drugs. Cutting a long story short, at some point I needed to calculate the right asymptote of this function: A(t) = Ln(k⋅t) - k⋅t where k,t ∈ℝ+. The derivative of A(t) tends to -k for t→∞, so I thought...
  24. 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...
  25. L

    Definition of an Asymptote

    Hello, i'm having some trouble understanding the definition of an asymptote, or rather the conditions that must be met in order for a line to be one. I have; "Let f : A \longrightarrow B be a function and A \subset R, B \subset R. A straight line is called an asymptote if one of the following...
  26. A

    MHB Equation for the horizontal asymptote.

    How would you find the equation for the horizontal asymptote of the following exponential function?: g(x) = -2f(2x - 6) Let f(x) = 4^x
  27. I

    MHB Finding Slant/Oblique Asymptote

    Hey! I know how to find slant asymptotes of regular rational functions, but what happens when the function is $y= \frac{x}{2} - \tan^{-1}x$ ? Is there a special way to do this? I know what the $\arctan x$ function looks like and that is $y\in(-\frac{\pi}{2},\,\frac{\pi}{2})$ and it is...
  28. C

    Linear oblique asymptote

    Homework Statement what is the linear oblique asymptote of (x^5+x^3+2)/(x^4-1) ? Homework Equations x-a/p(x) = q(x) +remainder The Attempt at a Solution I put in all the placeholders for the divisor and the numerator and got x as the equation for the linear oblique asymptote?? Is that right??
  29. U

    Construct ODE that approaches an asymptote

    Homework Statement Construct a first order linear differential equation whose solutions have the required behavior as t approaches infinity. Then solve your equation and confirm that the solutions do indeed have the specified property. All solutions are asymptotic to the line y = 2 - t as t...
  30. M

    MHB Why is this an asymptote?

    Why is the slant asymptote pictured here correct for this function? I was under the impression an asymptote was never crossed by the function. I get that the dividend gives the equation for the asymptote for a non zero remainder, but seeing this graphically is a bit confusing. Thanks! (EDIT...
  31. Petrus

    MHB Horizontal Asymptote of Inverse Tangent Function

    Hello MHB, I got one question, I am currently working with an old exam and I am suposed to draw it with vertican/horizontal lines (and those that are oblique). f(x)=\frac{x}{2}+\tan^{-1}(\frac{1}{x}) for the horizontel line \lim_{x->\infty^{\pm}}\frac{x}{2}+\tan^{-1}(\frac{x}{2}) Is it enough...
  32. G

    Asymptote vs hole in graph

    I don't understand why there is a hole in the graph of a function when there is a non-zero number in the numerator of a function and zero in the denominator, but an asymptote when both the numerator and the denominator are zeroes. Can someone explain why this is the case?
  33. P

    Horizontal Asymptote by Long Division?

    Homework Statement Find the horizontal asymptote Homework Equations The Attempt at a Solution There are two ways I did this problem. One way seems to be a coincidence and the other the "proper way" 1. I used long division, and got 2+(9x-3)/(x^2-2x) 2 seems to be the horizontal asymptote...
  34. Z

    Calculating asymptote of the function

    Homework Statement I would like to find an asymptote of the following function: f(x) = \sqrt{\frac{x^3}{x+1}} + x as x goes to negative infinity. 2. The attempt at a solution I calculated the limit of the function as x goes to -∞ which is ∞. However, this is not enough for me. I would like...
  35. I

    Getting equation from graph of a rational function with an oblique asymptote.

    Homework Statement -I have the zero, which is x=-1, however its a squared zero {(x+1)^2)} -Vertical asymptote is at x=1 -Equation of oblique asymptote is y=x+4 Homework Equations The Attempt at a Solution I tried finding the numerator by multiplying the oblique asymptote by the...
  36. P

    Horizontal Asymptote Question

    This is just a general concept question. Why is that an equation with a numerator to a greater degree than the denominator has no asymptote & the opposite does? Also why is the coefficient of the variable to the highest degree the horizontal asymptote? Maybe a proof would help here :P Anyway...
  37. D

    Limits question and finding oblique asymptote

    Question: Guess the oblique asymptote of the graph f(x) for x→∞. Write down the limit you have to compute to prove that your guess is correct. f(x)= \sqrt{(x^{4}+1)/(x^{2}-1)} so the limit would be: lim x→∞ \sqrt{(x^{4}+1)/(x^{2}-1)} I sketched out a graph but I just have no clue how to...
  38. D

    Trig: Writing the equation for vertical asymptote of a secant function?

    Homework Statement Homework Equations How did they come up with \frac{1}{2}+k for the equation of the vertical asymptote? I understand everything else except this.The Attempt at a Solution On this particular exercise, I graphed it and saw that each of my vertical dashed lines were all one whole...
  39. J

    Limit at infinity, no vert asymptote, why?

    Homework Statement Find the vertical asymptote(n) and evaluate the limit as x \rightarrow n^-, x\rightarrow n^+, or state Does Not Exist. Homework Equations \frac{\sqrt{4x^2+2x+10}-4}{x-1} The Attempt at a Solution I have taken the limits at \pm\infty=2,-2 and understand those are my...
  40. V

    Asymptotes and Hyperbolas: Exploring the Relationship

    Homework Statement If a graph has an asymptote, does that mean it's always going to be a hyperbola? Homework Equations The Attempt at a Solution Well, I started to think of y=tan(x) and y=cot(x). I believe they would be called trigonometric circular functions as they repeat, but...
  41. M

    Why does this expression not have a vertical asymptote?

    Homework Statement Find the vertical asymptote, if there is one, of this rational function. \frac{\sqrt{16x^2 + 3x + 6} -5}{x-1} Homework Equations The Attempt at a Solution This was actually a Calculus problem, I had to find the limit at infinity. I was able to do that easily; an extra...
  42. G

    Finding Horizontal Asymptote Process

    Homework Statement Find the horizontal asymptote of f(x) = (x2-1)/(x2-4) Homework Equations Limits The Attempt at a Solution I'm pretty family with the process, I just get confused when there's an x2 in the denominator and no x in the numerator ): So I already figured out that...
  43. G

    Find the vertical asymptote

    Homework Statement Sketch the graphs of f(x) = (x^3)/(x^2-1) showing vertical and horizontal asymptotes and relative extrema Homework Equations Zeroes, limits The Attempt at a Solution I've actually figured out the question; No horizontal asymptote, max at (-sqrt(3)...
  44. dkotschessaa

    Vertical Asymptote: Is f Defined at x=1?

    Homework Statement True False If the line x=1 is a vertical asymptote of y = f(x), then f is not defined at 1.Homework Equations none The Attempt at a Solution I originally believed this was true, but on finding it was false it sought a counter example: if for example f(x) = 1/x if x !=...
  45. J

    Is there a horizontal asymptote for y = 6/x - 3

    Homework Statement would there be a horizontal asymptote for y= 6/x - 3 Homework Equations I know that the vertical asymptote is x =3 because there the expression is undefined The Attempt at a Solution
  46. T

    Asymptote of curves problems

    Homework Statement Find the set of values of m such that the asymtote of the curve, y=\frac{3(m+1)x+m-2}{(m-2)x+3m} intersect at a point above the line y=3x-2 Homework Equations The Attempt at a Solution Vertical asymtote, x=-3m/(m-2) horizontal asymtote, y=3(m+1)/(m-2)...
  47. I

    Find the horizontal asymptote

    Homework Statement Find the horizontal asymptote(if there is one) using the rule for determining the horizontal asymptote of a rational function for (x^2+x-12)/ (x^2 -4)Homework Equations The Attempt at a Solution the degree of the numerator and denominator are both 2. Y=(An)/(Bn) Y=1/1 Y=1...
  48. Buckethead

    Asymptote of expected rotation curve velocities

    Can someone tell me why the expected velocity of the outer most stars of spiral galaxies has an asymptote quite a bit greater than zero? For example NGC 3198 at a radius of 50 kpc appears to be reaching it's asymptote at about 40-50 km/s which seems illogical.
  49. L

    Vertical asymptote of cos graph?

    Homework Statement State one known vertical asymptote of the graph of 1/secx Homework Equations None The Attempt at a Solution 1/secx = cosx
  50. L

    Finding Vertical Asymptote with limits

    How do I find a vertical asymptote of a function by using limits? I can find the HA by taking the limit, but how do I get the VA? What if the denominator is a square root? FIGURED OUT: I need to evaluate the limit as x approaches a from the right and left.