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Horizontal Asymptote Rules and Defination

Horizontal Asymptote Rules

Horizontal Asymptote Rules | Defination

Horizontal Asymptote Rules: In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) 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. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. 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.

Horizontal Asymptote Rules
Horizontal Asymptote Rules

There are three kinds of asymptotes: horizontal, vertical and oblique asymptotes. 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 −∞.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that tells you how the function will behave at the very edges of a graph. A horizontal asymptote is not sacred ground, however. The function can touch and even cross over the asymptote.

Horizontal asymptotes exist for functions where both the numerator and denominator are polynomials. These functions are called rational expressions. Let’s look at one to see what a horizontal asymptote looks like.

So, our function is a fraction of two polynomials. Our horizontal asymptote is y = 0. Look at how the function’s graph gets closer and closer to that line as it approaches the ends of the graph. We can plot some points to see how the function behaves at the very far ends.

x y
-10,000 -0.0004
-1000 -0.004
-100 -0.04
-10 -0.4
-1 -4
1 4
10 0.4
100 0.04
1000 0.004
10,000 0.0004

Do you see how the function gets closer and closer to the line y = 0 at the very far edges? This is how a function behaves around its horizontal asymptote if it has one. Not all rational expressions have horizontal asymptotes. Let’s talk about the rules of horizontal asymptotes now to see in what cases a horizontal asymptote will exist and how it will behave.

Horizontal Asymptote Examles

  • f(x)=4*x^2-5*x / x^2-2*x+1

First we must compare the degrees of the polynomials. Both the numerator and denominator are 2nd degree polynomials. Since they are the same degree, we must divide the coefficients of the highest terms.

In the numerator, the coefficient of the highest term is 4.

In the denominator, the coefficient of the highest term is an understood 1.

4/1=4

The horizontal asymptote is at y = 4.

  • f(x)=x^2-9 / x+10

First we must compare the degrees of the polynomials. The numerator contains a 2nddegree polynomial while the denominator contains a 1st degree polynomial.

Since the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote instead.

Vertical Asymptotes

An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.

If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd.

Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.

Oblique Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. Another name for an oblique asymptote is a slant asymptote.

To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator. As x gets very large (this is the far left or far right that I was talking about), the remainder portion becomes very small, almost zero. So, to find the equation of the oblique asymptote, perform the long division and discard the remainder.

Oblique Asymptotes
Oblique Asymptotes

Curvilinear Asymptotes

Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred to as an asymptote of A, when there is no risk of confusion with linear asymptotes.

For example, the function

has a curvilinear asymptote y = x2 + 2x + 3, which is known as a parabolic asymptote because it is a parabola rather than a straight line.

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