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.
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...
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...
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?
$\tiny{s8.3.6.46}$
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...
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...
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.
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...
[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...
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...
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?
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.
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**
$$\ r\theta =1$$
$$\...
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...
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
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...
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 ∞...
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...
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...
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...
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...
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...
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??
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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 !=...
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
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)...
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...
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.
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.