Horizontal Asymptote Rules And Definition

Horizontal Asymptote Rules And Definition

Horizontal Asymptote Rules: Before getting into the definition of a horizontal asymptote, let’s first go over what a function is. A function is an equation that tells you how two things relate. Usually, functions tell you how y is related to x. Functions are often graphed to provide a visual.

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 Asymptote Rules And Definition

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.

Rules For Horizontal Asymptote

Rules For Horizontal Asymptote

Whereas vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions.

Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes.

Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior, usually far off to the sides of the graph.

In other words, horizontal asymptotes are different from vertical asymptotes in some fairly significant ways.

To understand the concept of horizontal asymptotes, let’s look at a few examples.

  • Find the horizontal asymptote of the following function:

\mathbf{\color{green}{\mathit{y} = \dfrac{\mathit{x} + 2}{\mathit{x}^2 + 1}}}

First, notice that the denominator is a sum of squares, so it doesn’t factor and has no real zeroes. In other words, this rational function has no vertical asymptotes. So we’re okay on that front.

As mentioned above, the horizontal asymptote of a function (assuming it has one) tells me roughly where the graph will be going when x gets really, really big. So I’ll look at some very big values for x; that is, at some values of x which are very far from the origin:

x

y = \dfrac{x + 2}{x^2 + 1}

–100 000

–0.0000099…

–10 000

–0.0000999…

–1 000

–0.0009979…

–100

–0.0097990…

–10

–0.0792079…

–1

0.5

0

2

1

1.5

10

0.1188118…

100

0.0101989…

1 000

0.0010019…

10 000

0.0001000…

100 000

0.0000100…

Off to the sides of the graph, where x is strongly negative (such as –1,000) or else strongly positive (such as 10000), the “+2” and the “+1” in the expression for y really don’t matter so much. I ended up having a really big number divided by a really big number squared, which “simplified” to be a very small number. The values of y came mostly from the “x” and the “x2“, especially once x got very large. And since the x2 was “bigger” than the x, the x2 dragged the value of the whole fraction down to y = 0 (that is, down to the x-axis) when x got big.

This makes perfect sense when you think about it. If you’ve got a zillion (plus two, but who cares about that?) divided by a zillion squared (plus 1, but who cares about that?), then you’ve essentially got a zillion divided by the square of a zillion, which simplifies to 1 over a zillion. Which is very, very small. So, of course, the value of the function gets very, very small; namely, it gets very, very close to zero.

Horizontal Asymptote Degree Rules

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 they 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 Asymptotes Rules

There are three rules that horizontal asymptotes follow depending on the degree of the polynomials involved in the rational expression. Before we begin, let’s define our function like this:

horizontal asymptote

Our function has a polynomial of degree n on top and a polynomial of degree m on the bottom. Our horizontal asymptote rules are based on these degrees.

  1. When n is less than m, the horizontal asymptote is y = 0 or the x-axis.
  2. When n is equal to m, then the horizontal asymptote is equal to y = a/b.
  3. When n is greater than m, there is no horizontal asymptote.

The degrees of the polynomials in the function determine whether there is a horizontal asymptote and where it will be. Let’s see how we can use these rules to figure out horizontal asymptotes

Rules For Finding Horizontal Asymptote

Remember that an asymptote is a line that the graph of a function approaches but never touches. Rational functions contain asymptotes, as seen in this example:

In this example, there is a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The curves approach these asymptotes but never cross them.

The method used to find the horizontal asymptote changes depending on how the degrees of the polynomials in the numerator and denominator of the function compare.

    • If both polynomials are the same degree, divide the coefficients of the highest degree terms.
      Example:
      Both polynomials are 2nd degree, so the asymptote is at 
    • If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.
  • If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote, which we will study in a later lesson

Examples: Find the horizontal asymptote of each rational function:

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 understood 1.
The horizontal asymptote is at y = 4.
First, we must compare the degrees of the polynomials. The numerator contains the 2nd-degree polynomial while the denominator contains the 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.
First, we must compare the degrees of the polynomials. The numerator contains the 1st-degree polynomial while the denominator contains the 3rd-degree polynomial.
Since the polynomial in the numerator is a lower degree than the denominator, the horizontal asymptote is located at y=0.

Horizontal Asymptote Rules Rational Functions

If your function is rational, that is, if f(x) has the form of a fraction, f(x) = p(x) / q(x), in which both p(x) and q(x) are polynomials, then you can use highest order term analysis. The highest-order term of a polynomial p(x) is the single term having the greatest degree (exponent on x). For example, the highest order term of 6x – 3x5 + 5x3 + 42 is  –3x5.

Highest Order Term Analysis

To do the highest order term analysis on a rational function, make sure the top and bottom polynomials are fully expanded and then write a new function having only the highest order term from the top and from the bottom. All other terms (lower-order terms) can safely be ignored. Cancel any common factors and variables and:

  • If the result is a constant k, then y = k is the single horizontal asymptote. This happens when the degree of the top matches the degree of the bottom.

  • If the result has any powers of x leftover on top, then there is no horizontal asymptote.

  • If the result has any powers of x leftover on the bottom, then y = 0 is the single horizontal asymptote.

Examples for Highest Order Term Analysis

Let’s use the highest order term analysis to find the horizontal asymptotes of the following functions.

Three horizontal asymptote problems

(a) The highest-order term on the top is 6x2, and on the bottom, 3x2. Dividing and canceling, we get (6x2)/(3x2) = 2, a constant. Therefore the horizontal asymptote is y = 2.

(b) Highest order term analysis leads to (3x3)/(x5) = 3/x2, and since there are powers of x leftover on the bottom, the horizontal asymptote is automatically y = 0.

(c) This time, there are no horizontal asymptotes because (x4)/(x3) = x/1, leaving an x on the top of the fraction.

Which function has no horizontal asymptote?

A rational function has no horizontal asymptote when the degree of the numerator is greater than the denominator. A) f(x)=2x-1/3x^2, the degree of the numerator is lower than the denominator. B) f(x)=x-1/3x, the degree of the numerator is lower than the denominator.

Can a function have more than one horizontal asymptote?

The answer is no, a function cannot have more than two horizontal asymptotes.

Why can graphs cross horizontal asymptotes?

The graph crosses the x-axis at x=0. For x> 0, it rises to a maximum value and then decreases toward y= 0 as x goes to infinity. For x< 0, it decreases to a minimum value then rises toward y= 0 as x goes to negative infinity. y= 0 is a horizontal asymptote but the graph crosses y= 0 at x= 0.