An asymptote is a line that a graph approaches, but does not intersect.

In this lesson, we will learn how to find vertical asymptotes, horizontal asymptotes and oblique (slant) asymptotes of rational functions.

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More lessons on Calculus

** Method 1: Use the Definition of Vertical Asymptote **

The line *x* = *a* is called a Vertical Asymptote of the curve *y* = *f*(*x*) if at least one of the following statements is true.

**Method 2: **

For rational functions, vertical asymptotes are vertical lines that correspond to the zeroes of the denominator.

Given the rational function, *f*(*x*)

* Step 1:* Write

* Step 2:* if

** Example:**

Find the vertical asymptotes of

** Solution: **

**Method 1:** Use the definition of Vertical Asymptote.

If *x* is close to 3 but larger than 3, then the denominator *x* – 3 is a small positive number and 2*x* is close to 8. So, is a large positive number.

Intuitively, we see that

Similarly, if *x* is close to 3 but smaller than 3, then *x* – 3 is a small negative number and 2*x* is close to 8. So, is a large negative number.

The line *x* = 3 is the vertical asymptote.

**
Method 2:**

**Step 1:*** f*(*x*) is already in reduced form.

* Step 2:* The denominator is

** Method 1: Use the definition of Horizontal Asymptote **

The line *y* = *L* is called a horizontal asymptote of the curve *y* = *f*(*x*) if either

Method 2:

For the rational function, *f*(*x*)

If the degree of *x* in the numerator is less than the degree of *x* in the denominator then *y* = 0 is the horizontal asymptote.

If the degree of *x* in the numerator is equal to the degree of *x* in the denominator then *y* = *c* where *c* is obtained by dividing the leading coefficients.

** Example: **

Find the horizontal and vertical asymptotes of the function.

** Solution: **

** Method 1: **

Divide both numerator and denominator by x.

The line is the horizontal asymptote.

** Method 2: **

The degree of *x* in the numerator is equal to the degree of *x* in the denominator.

Dividing the leading coefficients we get

The line is the horizontal asymptote.

This video will give a basic overview of horizontal asymptotes. We will determine if the given rational functions have horizontal asymptotes and what they are.

This video will go into further detail about horizontal asymptote rules.

Some curves have asymptotes that are oblique, that is, neither horizontal nor vertical.

If then the line *y* = *mx* + *b* is called the oblique or slant asymptote because the vertical distances between the curve *y* = *f*(*x*) and the line *y* = *mx* + *b* approaches 0.

For rational functions, oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the oblique asymptote can be found by long division.

** Example: **

Find the asymptotes of the function

** Solution: **

Since the denominator *x*^{2 }+ 1 is never 0, there is no vertical asymptote.

Since the degree of *x* in the numerator is greater than the degree of *x* in the denominator there is no horizontal asymptote.

Since the degree of *x* in the numerator is one greater than the degree of *x* in the denominator we can use long division to obtain the oblique asymptote.

So, the line *y* = *x* is the oblique asymptote.