#### Horizontal Asymptotes Rules: Definition and Examples

In order to understand what a horizontal asymptote is, let’s first consider what a function is. A function is an equation that describes the relationship between two things. The function describes the relationship between x and y. Functions can be graphed in order to provide a visual representation.

Asymptotes are lines that show how a function behaves at the very edges of a graph. However, horizontal asymptotes are not inviolable. It is possible for the function to touch and even cross over the asymptote.

For functions with polynomials in both the numerator and denominator, horizontal asymptotes exist. This is known as a rational expression. Check out this horizontal asymptote to see what it looks like.

Therefore, our function is a fraction of two polynomials. Our horizontal asymptote is y = 0. As you approach the ends of the graph, you can see that the function’s graph gets closer and closer to the line. Plotting some points will help us see how the function behaves at the very far ends.

## Horizontal Asymptote Degree Rules

### Horizontal Asymptote Degree

A graph’s asymptote horizontal is a line that shows how the function behaves at the extreme edge. However, an asymptote horizontal is not sacred ground. In some cases, the function will even touch or cross the asymptote.

It is possible to get a horizontal asymptote for functions with polynomials in both the numerator and denominator. This type of function is called a rational expression. You can see an asymptote horizontally by looking at one.

Therefore, our function is a fraction of two polynomials. Our horizontal asymptote is y = 0. As the graph approaches the ends of the graph, the function’s graph gets closer and closer to that line. By plotting some points, we can see how their 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

Isn’t it interesting how the function gets closer and closer to the line y = 0 at the very edges? A function behaves in this way around its horizontal asymptote if it has one. However, not all rational expressions possess horizontal asymptotes. We will now look at the rules of horizontal asymptotes to see in what cases it will exist and how they will behave.

### Horizontal Asymptote Examples

f(x)=4*x^2-5*x / x^2-2*x+1

The degree of each polynomial must be compared first. A 2nd-degree polynomial is both the numerator and denominator. The coefficients of the highest terms must be divided since they have the same degree.

There is a coefficient of 4 for the highest term in the numerator.

One is taken for the coefficient of the highest term in the denominator.

4/1=4

At y = 4, the horizontal asymptote is reached.

f(x)=x^2-9 / x+10

The degree of each polynomial must first be compared. The numerator includes a polynomial of the second degree, and the denominator includes a polynomial of the first degree.

The polynomial in the numerator is of a higher degree than the polynomial in the denominator, so there is no horizontal asymptote. Instead, there is a slant asymptote.

## FAQs

Does it make sense for a function to have more than one horizontal asymptote?

Essentially, a function can have a maximum of two horizontal asymptotes.

What causes graphs to cross horizontal asymptotes?

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

What is the procedure for finding vertical and horizontal asymptotes?

The graph will have a vertical asymptote at x = 1 because the denominator is zero: x * 1 = 0 x = 1 Therefore, the graph will have a vertical asymptote at x = 1. In order to determine the horizontal asymptote, we note that the degree of the numerator is two and the degree of the denominator is one.

Is it possible for a graph to cross a horizontal asymptote?

It is a common misconception among students that a graph cannot cross a slant or horizontal asymptote. There can be multiple crossings of horizontal and slant asymptotes in a graph (sometimes more than once). There are some vertical asymptotes that a graph can’t cross. These are the bad points of the domain.

When solving an equation, how do you find the asymptote?

Assume n(x) = 0, where n(x) is the denominator of the function (note: this only applies if t(x) is not zero for the same x value). The graph shows an asymptote with the equation x = 1.

Which function does not have a horizontal asymptote?

A rational function has no horizontal asymptote when the degree of the numerator exceeds the degree of the denominator. The degree of the numerator is lower than the denominator in f(x)=2x-1/3x*2. In case f(x)=x-1/3x, the degree of the numerator is lower than that of the denominator.

What is the procedure for finding the vertical asymptote of a function?

Set the denominator equal to 0 and solve for x to find the vertical asymptotes of a rational function.

Asymptotes in exponential functions are horizontal. What does this mean?

Asymptotes may be vertical, oblique, or horizontal. As x becomes very large or very small, the curve approaches the horizontal asymptote. It is defined as any function in which the independent variable is expressed as an exponent; it is the inverse of a logarithm.

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