# HOW TO FIND SLANT ASYMPTOTE OF A FUNCTION

## About the topic "How to find slant asymptote of a function"

"How to find slant 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 slant 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 slant asymptote of a function.

And we will be able to find slant 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

## 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, or if the highest exponent of the numerator is less than the highest exponent of the denominator, there is no slant asymptote.

Case 2 :

If the highest exponent of the numerator is greater than the highest exponent of the denominator by one, there is a slant asymptote.

To find slant asymptote, we have to use long division to divide the numerator by denominator. When we divide so, let the quotient be (ax+b).

Then, the equation of the slant asymptote is y = ax + b

## 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, there is no slant asymptote.

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.

Hence, there is no slant asymptote.

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

f(x) = (x² +3x+2)/(x-2)

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 by one. So, there is a slant asymptote.

Step 3 :

To get the equation of the slant asymptote, we have to divide the numerator by the denominator using long division as given below.

Step 3 :

In the above long division, the quotient is (x+5).

Hence, the equation of the slant asymptote is y = x+5