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# 12.6: Determining Asymptotes by Division

Difficulty Level: At Grade Created by: CK-12
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Practice Determining Asymptotes by Division

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### Determining Asymptotes by Division

In the last section we saw how to find vertical and horizontal asymptotes. Remember, the horizontal asymptote shows the value of \begin{align*}y\end{align*} that the function approaches for large values of \begin{align*}x\end{align*}. 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.

#### Finding Asymptotes

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

Find the horizontal asymptote of \begin{align*}y=\frac{2}{x-1}\end{align*}.

We can’t reduce this fraction, and as \begin{align*}x\end{align*} gets larger the denominator of the fraction gets much bigger than the numerator, so the whole fraction approaches zero.

The horizontal asymptote is \begin{align*}y = 0\end{align*}.

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

Find the horizontal asymptote of \begin{align*}y=\frac{3x+2}{x-1}\end{align*}.

In this case we can divide the two polynomials:

\begin{align*}& \overset{\qquad \qquad \ 3}{x-1 \overline{ ) 3x+2 \;}}\\ & \qquad \underline{-3x+3}\\ & \qquad \qquad \quad 5\end{align*}

So the expression can be written as \begin{align*}y=3+\frac{5}{x-1}\end{align*}.

Because the denominator of the remainder is bigger than the numerator of the remainder, the remainder will approach zero for large values of \begin{align*}x\end{align*}. Adding the 3 to that 0 means the whole expression will approach 3.

The horizontal asymptote is \begin{align*}y = 3\end{align*}.

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

Find any asymptotes of \begin{align*}y=\frac{4x^2+3x+2}{x-1}\end{align*}.

Solution:

We can do long division once again and rewrite the expression as \begin{align*}y=4x+7+\frac{9}{x-1}\end{align*}. The fraction here approaches zero for large values of \begin{align*}x\end{align*}, so the whole expression approaches \begin{align*}4x + 7\end{align*}.

When the rational function approaches a straight line for large values of \begin{align*}x\end{align*}, we say that the rational function has an oblique asymptote. In this case, then, the oblique asymptote is \begin{align*}y = 4x + 7\end{align*}.

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

Find any asymptotes of \begin{align*}y=\frac{x^3}{x-1}\end{align*}.

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.

### Examples

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

#### Example 1

\begin{align*}y=\frac{3x^2}{x^2+4}\end{align*}

When we simplify the function, we get \begin{align*}y=3-\frac{12}{x^2+4}\end{align*}. There is a horizontal asymptote at \begin{align*}y = 3\end{align*}.

#### Example 2

\begin{align*}y=\frac{x-1}{3x^2-6}\end{align*}

We cannot divide the two polynomials. There is a horizontal asymptote at \begin{align*}y = 0\end{align*}.

#### Example 3

\begin{align*}y=\frac{x^4+1}{x-5}\end{align*}

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

#### Example 4

\begin{align*}y=\frac{x^3-3x^2+4x-1}{x^2-2}\end{align*}

When we simplify the function, we get \begin{align*}y=x-3+\frac{6x-7}{x^2-2}\end{align*}. There is an oblique asymptote at \begin{align*}y = x - 3\end{align*}.

### Review

Find all asymptotes of the following rational functions:

1. \begin{align*}\frac{x^2}{x-2}\end{align*}
2. \begin{align*}\frac{1}{x+4}\end{align*}
3. \begin{align*}\frac{x^2-1}{x^2+1}\end{align*}
4. \begin{align*}\frac{x-4}{x^2-9}\end{align*}
5. \begin{align*}\frac{x^2+2x+1}{4x-1}\end{align*}
6. \begin{align*}\frac{x^3+1}{4x-1}\end{align*}
7. \begin{align*}\frac{x-x^3}{x^2-6x-7}\end{align*}
8. \begin{align*}\frac{x^4-2x}{8x+24}\end{align*}

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

1. \begin{align*}\frac{x^2}{x+2}\end{align*}
2. \begin{align*}\frac{x^3-1}{x^2-4}\end{align*}
3. \begin{align*}\frac{x^2+1}{2x-4}\end{align*}
4. \begin{align*}\frac{x-x^2}{3x+2}\end{align*}

To view the Review answers, open this PDF file and look for section 12.6.

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### Vocabulary Language: English

TermDefinition
Function A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Horizontal Asymptote A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.
Ohm's Law Ohm's Law states that a current through a conductor that connects two points is directly proportional to the potential difference between its ends. Consider $V=IR$, where $V$ is the voltage or the potential difference, $I$ is the current, and $R$ is the resistance of the conductor.

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Date Created:
Oct 01, 2012
Apr 11, 2016
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MAT.ALG.766.3.L.1