Finding Horizontal Asymptotes in Rational Functions

Oct 13, 2025 by windailys.com 148 views

Introduction

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. Rational functions, which are the ratio of two polynomials, can have horizontal asymptotes. In this article, we will explore how to find the horizontal asymptote of a rational function, specifically the function $j(x)=\frac{14 x^3+45}{2 x^3+x+9}$ .

What is a Horizontal Asymptote?

A horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. In other words, it is a line that the function gets arbitrarily close to as x goes to positive or negative infinity. Horizontal asymptotes are important in mathematics because they help us understand the behavior of a function as x gets very large or very small.

Finding Horizontal Asymptotes in Rational Functions

To find the horizontal asymptote of a rational function, we need to look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 1: Determine the Degrees of the Numerator and Denominator

The numerator of the function $j(x)=\frac{14 x^3+45}{2 x^3+x+9}$ is $14 x^3+45$ , which has a degree of 3. The denominator is $2 x^3+x+9$ , which also has a degree of 3.

Step 2: Determine the Leading Coefficients

The leading coefficient of the numerator is 14, and the leading coefficient of the denominator is 2.

Step 3: Find the Horizontal Asymptote

Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. Therefore, the horizontal asymptote of the function $j(x)=\frac{14 x^3+45}{2 x^3+x+9}$ is $y=\frac{14}{2}=7$ .

Conclusion

In this article, we explored how to find the horizontal asymptote of a rational function. We used the function $j(x)=\frac{14 x^3+45}{2 x^3+x+9}$ as an example and determined that the horizontal asymptote is $y=7$ . We also discussed the importance of horizontal asymptotes in mathematics and how they help us understand the behavior of a function as x gets very large or very small.

Answer

The correct answer is C. $y=7$ .

Additional Examples

Here are a few more examples of finding horizontal asymptotes in rational functions:

$f(x)=\frac{3 x^2+2}{x^2-4}$ : The degree of the numerator is equal to the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients. Therefore, the horizontal asymptote of the function $f(x)=\frac{3 x^2+2}{x^2-4}$ is $y=\frac{3}{1}=3$ .
$g(x)=\frac{x^3+2}{x^3-1}$ : The degree of the numerator is equal to the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients. Therefore, the horizontal asymptote of the function $g(x)=\frac{x^3+2}{x^3-1}$ is $y=\frac{1}{1}=1$ .
$h(x)=\frac{2 x^2+3}{x^2-2}$ : The degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is y=0. Therefore, the horizontal asymptote of the function $h(x)=\frac{2 x^2+3}{x^2-2}$ is $y=0$ .

Final Thoughts

Frequently Asked Questions

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. In other words, it is a line that the function gets arbitrarily close to as x goes to positive or negative infinity.

Q: How do I find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: What if the degree of the numerator is equal to the degree of the denominator?

A: If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. For example, if the numerator is $3 x^2+2$ and the denominator is $x^2-4$ , the horizontal asymptote is $y=\frac{3}{1}=3$ .

Q: What if the degree of the numerator is greater than the degree of the denominator?

A: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. For example, if the numerator is $3 x^3+2$ and the denominator is $x^2-4$ , there is no horizontal asymptote.

Q: Can a rational function have more than one horizontal asymptote?

A: No, a rational function can only have one horizontal asymptote. If a rational function has a horizontal asymptote, it is the only horizontal asymptote.

Q: How do I determine if a rational function has a horizontal asymptote?

A: To determine if a rational function has a horizontal asymptote, you need to look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote. If the degree of the numerator is equal to the degree of the denominator, the function has a horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote.

Q: Can a rational function have a slant asymptote?

A: Yes, a rational function can have a slant asymptote. A slant asymptote is a line that a function approaches as the absolute value of the x-coordinate gets larger and larger, but it is not a horizontal line.

Q: How do I find the slant asymptote of a rational function?

A: To find the slant asymptote of a rational function, you need to divide the numerator by the denominator using long division or synthetic division. The slant asymptote is the quotient of the division.

Q: Can a rational function have both a horizontal and a slant asymptote?

A: No, a rational function can only have one type of asymptote. If a rational function has a horizontal asymptote, it does not have a slant asymptote. If a rational function has a slant asymptote, it does not have a horizontal asymptote.

Conclusion

In conclusion, finding horizontal asymptotes in rational functions is an important concept in mathematics. By following the steps outlined in this article, you can determine the horizontal asymptote of a rational function and understand the behavior of the function as x gets very large or very small.