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 :

In this section, we are going to learn, how to find horizontal asymptotes of a function. .  

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

That is, the function has to be in the form of

f(x)  =  g(x) / h(x)

Example : Rational Function

Steps to Find Horizontal Asymptotes of a Rational Function

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. 

How to Find Horizontal Asymptote of a Function - Examples

Example 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

Example 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

Example 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.

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Horizontal Asymptote of a Function". 

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