Horizontal Asymptote Rules 

Mathematics offers interesting insights when dealing with graphs. Asymptotes are among them, offering deeper insights into the nature of functions. They appear to approach the axis but do not, making them worthy of understanding. As they are a significant part of the syllabus and foundation for the competitive and preparatory exams, clarity on the concepts is essential. Go through the article to get aid in solving the question while understanding the associated intricacies. 

What is an Asymptote? 

Represented by dashed or dotted lines on a graph, the asymptote is a straight line that a curve approaches but never meets. Alternatively, the line and curve can be said to meet at an infinite distance. There are three types of asymptotes: horizontal, vertical, and oblique or slant asymptotes. 

The horizontal asymptote refers to the behaviour of a curve or line as the independent variable x approaches positive or negative infinity. It is represented by the horizontal dashed lines parallel to the x-axis. A vertical asymptote is marked by the vertical dashed lines parallel to the y-axis. It represents the function’s restricted values for x. Oblique asymptote is the slant asymptote exhibited by the linear equation of the form y=mx+b. It is diagonal across the graph in nature. 

What Are the Rules of Horizontal Asymptote?

Three rules of horizontal asymptote are applicable when the function is given. The rules are based on the degree of polynomials of the numerator and denominator. In case of absence of function, the horizontal asymptote needs to be calculated. Going back to the rules, they are as follows:  

Case 1: Equal degree of polynomial 

If the degree of a polynomial of the numerator and denominator is equal, then the division of the coefficients of the highest degree terms will decide the horizontal asymptote. The resultant divided value will be the horizontal asymptote. 

Case 2: Lower degree in the numerator 

In this case, the horizontal asymptote will be y=0. 

Case 3: Greater degree in the numerator 

In the case of a higher-degree numerator, there will not be a horizontal asymptote. 

Case 4: Exponential Function

The exponential function will always have only one horizontal asymptote. Further, depending on the form of the equation, the horizontal asymptote will be either 0 or a constant value. 

Examples of Horizontal Asymptote 

You can be given any type of expression to find the horizontal asymptote. Learn from the distinct types of examples here. 

Finding the Horizontal Asymptote From the Rational Equation

Let us say you are presented with the following question. 

Q1. Find the horizontal asymptote of the function g(y)=3y2+1/4y2-2. 

Solution: You can use the limit of the function to solve it. Here is how it will go: 

y=3y2+1/4y2-2

y=(3y2+1)(4y2+2)/(4y2-2) (4y2+2)

y=12y4 +10y2+2)/16y4-4

The higher orders will now be used for calculation as y approaches positive infinity. 

12y4 /16y4=12/16=3/4

Hence, the horizontal asymptote is y=3/4.

Q2. Find the horizontal asymptote of the g(x)=3×2/4.

Solution: In this function, the numerator has a greater degree of polynomial. Hence, there will be no horizontal asymptote. 

Q3. Find the horizontal asymptote of the g(x)=4/3×2.

Solution: As the denominator has a higher degree polynomial, the horizontal asymptote will be 0. 

Q4. Find the horizontal asymptote of the g(x)=3×2+2/3×2-2.

Solution: In the present case, we have the same degree of polynomials in both the numerator and denominator. This can be solved using the limit, as in question 1, or through the following method. 

Here, we will be dividing the highest coefficient, which is 3. Since the coefficients are the same for both the numerator and denominator, the resultant will be 1. Therefore, the horizontal asymptote for the function g(x)=3×2+2/3×2-2 is 1. 

Finding the Horizontal Asymptote in the Exponential Equation 

Q5. What is the horizontal asymptote of the following equations: 

Equation 1: 7x+9

Equation 2: 3x

Solution: 

Equation 1: The equation is the form: abkx+c. Here, y will be c. As 9 is the c or constant here, y = 9. Therefore, the horizontal asymptote is 9. 

Equation 2: The equation is the form: bx. Thus, the horizontal asymptote here will be 0. 

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