HOW TO FIND HORIZONTAL ASYMPTOTE OF A FUNCTION

About the topic "How to find horizontal asymptote of a function"

"How to find horizontal asymptote of a function ?" is the question having had by the students who are studying math in school final. Even though it is taught by the teachers in school and university, students do not understand this clearly. Often students have this question on horizontal asymptotes. 

On this page of our website, we have given step by step explanation and examples to make the students to clearly understand how to find horizontal asymptote of a function.  

And we will be able to find horizontal asymptote of a function, only if it is a rational function.

That is, the function has to be in the form of f(x) = P/Q

Example : Rational Function

Steps involved in finding horizontal asymptotes

Let f(x) be the given rational function. Compare the highest exponent of the numerator and denominator.

Case 1 :

If the highest exponent of the numerator and denominator are equal, equation of horizontal asymptote is

                                 y = a/b

Here "a" and "b" are the coefficients of highest exponent terms at the numerator and denominator respectively.  

Case 2 :

If the highest exponent of the numerator is less than the highest exponent of the denominator, equation of horizontal asymptote is

                                 y = o (or) x-axis

Case 3 :

If the highest exponent of the numerator is greater than the highest exponent of the denominator, there is no horizontal asymptote and there is only slant asymptote or oblique. 

Examples:

1. Find the equation of horizontal asymptote for the function given below. 

f(x) = 1/(x+6)

Solution :

Step 1:

In the given rational function, the highest exponent of the numerator is 0 and the highest exponent of the denominator is 1. 

Step 2 :

Clearly highest exponent of the numerator is less than the highest exponent of the denominator. 

Hence, equation of the horizontal asymptote is y = 0 (or) x-axis.

2. Find the equation of vertical asymptote for the function given below. 

f(x) = (x²+2x-3)/(x²-5x+6)

Solution :

Step 1:

In the given rational function, the highest exponent of the numerator is 2 and the highest exponent of the denominator is 2. 

Step 2 :

Clearly, the exponent of the numerator and the denominator are equal. 

Step 3 :

Now, to get the equation of the horizontal asymptote, we have to divide the coefficients of highest exponent terms of the numerator and denominator.

Hence, equation of the horizontal asymptote is y = 1/1

                                                                                 y = 1

3. Find the equation of vertical asymptote for the function given below. 

f(x) = (x² -4)/(2x-3)

Solution :

Step 1:

In the given rational function, the highest exponent of the numerator is 2 and the highest exponent of the denominator is 1.

Step 2 :

Clearly, the exponent of the numerator is greater than the exponent of the denominator. 

Step 3 :

As per the steps explained above, there is no horizontal asymptote.

You can also visit the following web pages related to "Horizontal Asymptotes"

http://www.purplemath.com

http://www.coolmath.com

http://www.softschools.com

https://www.khanacademy.org

http://www.freemathhelp.com




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