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.
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.
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:
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=x+x+ $

–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 “x^{2}“, especially once x got very large. And since the x^{2} was “bigger” than the x, the x^{2} dragged the value of the whole fraction down to y = 0 (that is, down to the xaxis) 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:
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.
 When n is less than m, the horizontal asymptote is y = 0 or the xaxis.
 When n is equal to m, then the horizontal asymptote is equal to y = a/b.
 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 2^{nd} degree, so the asymptote is at  If the polynomial in the numerator is a lower degree than the denominator, the xaxis (y = 0) is the horizontal asymptote.
 If both polynomials are the same degree, divide the coefficients of the highest degree terms.
 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:
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.
Since the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote instead.
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 highestorder 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 – 3x^{5} + 5x^{3} + 42 is –3x^{5}.
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 (lowerorder 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.
(a) The highestorder term on the top is 6x^{2}, and on the bottom, 3x^{2}. Dividing and canceling, we get (6x^{2})/(3x^{2}) = 2, a constant. Therefore the horizontal asymptote is y = 2.
(b) Highest order term analysis leads to (3x^{3})/(x^{5}) = 3/x^{2}, 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 (x^{4})/(x^{3}) = 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)=2x1/3x^2, the degree of the numerator is lower than the denominator. B) f(x)=x1/3x, the degree of the numerator is lower than the denominator.