What if you had a function like ? How could you rewrite it to find its asymptotes? After completing this Concept, you'll be able to rewrite rational functions like this one using division.

### Watch This

CK-12 Foundation: 1206S Rewriting Rational Functions Using Division

### Guidance

In the last section we saw how to find vertical and horizontal asymptotes. Remember, the horizontal asymptote shows the value of that the function approaches for large values of . Let’s review the method for finding horizontal asymptotes and see how it’s related to polynomial division.

When it comes to finding asymptotes, there are basically four different types of rational functions.

**Case 1:** The polynomial in the numerator has a lower degree than the polynomial in the denominator.

#### Example A

*Find the horizontal asymptote of .*

**Solution:**

We can’t reduce this fraction, and as gets larger the denominator of the fraction gets much bigger than the numerator, so the whole fraction approaches zero.

**The horizontal asymptote is .**

**Case 2:** The polynomial in the numerator has the same degree as the polynomial in the denominator.

#### Example B

*Find the horizontal asymptote of .*

**Solution:**

In this case we can divide the two polynomials:

So the expression can be written as .

Because the denominator of the remainder is bigger than the numerator of the remainder, the remainder will approach zero for large values of . Adding the 3 to that 0 means the whole expression will approach 3.

**The horizontal asymptote is .**

**Case 3:** The polynomial in the numerator has a degree that is one more than the polynomial in the denominator.

#### Example C

*Find any asymptotes of .*

**Solution:**

We can do long division once again and rewrite the expression as . The fraction here approaches zero for large values of , so the whole expression approaches .

When the rational function approaches a straight line for large values of , we say that the rational function has an **oblique asymptote.** In this case, then, **the oblique asymptote is .**

**Case 4:** The polynomial in the numerator has a degree that is two or more than the degree in the denominator.

#### Example D

*Find any asymptotes of .*

This is actually the simplest case of all: **the polynomial has no horizontal or oblique asymptotes.**

Notice that a rational function will either have a horizontal asymptote, an oblique asymptote or neither kind. In other words, a function can’t have both; in fact, it can’t have more than one of either kind. On the other hand, a rational function can have any number of *vertical* asymptotes at the same time that it has horizontal or oblique asymptotes.

Watch this video for help with the Examples above.

CK-12 Foundation: Rewriting Rational Functions Using Division

### Vocabulary

- When the rational function approaches a straight line for large values of , we say that the rational function has an
**oblique asymptote.**

### Guided Practice

*Find the horizontal or oblique asymptotes of the following rational functions.*

a)

b)

c)

d)

**Solution**

a) When we simplify the function, we get . **There is a horizontal asymptote at .**

b) We cannot divide the two polynomials. **There is a horizontal asymptote at .**

c) The power of the numerator is 3 more than the power of the denominator. **There are no horizontal or oblique asymptotes.**

d) When we simplify the function, we get . **There is an oblique asymptote at .**

### Practice

Find all asymptotes of the following rational functions:

Graph the following rational functions. Indicate all asymptotes on the graph: