12.6: Determining Asymptotes by Division
What if you had a function like \begin{align*}y=\frac{3x^2  2x + 1}{x + 2}\end{align*}
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CK12 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 \begin{align*}y\end{align*}
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 \begin{align*}y=\frac{2}{x1}\end{align*}
Solution:
We can’t reduce this fraction, and as \begin{align*}x\end{align*}
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.
Example B
Find the horizontal asymptote of \begin{align*}y=\frac{3x+2}{x1}\end{align*}
Solution:
In this case we can divide the two polynomials:
\begin{align*}& \overset{\qquad \qquad \ 3}{x1 \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}{x1}\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*}
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.
Example C
Find any asymptotes of \begin{align*}y=\frac{4x^2+3x+2}{x1}\end{align*}
Solution:
We can do long division once again and rewrite the expression as \begin{align*}y=4x+7+\frac{9}{x1}\end{align*}
When the rational function approaches a straight line for large values of \begin{align*}x\end{align*}
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 \begin{align*}y=\frac{x^3}{x1}\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.
Watch this video for help with the Examples above.
CK12 Foundation: Rewriting Rational Functions Using Division
Vocabulary
 When the rational function approaches a straight line for large values of \begin{align*}x\end{align*}
x , we say that the rational function has an oblique asymptote.
Guided Practice
Find the horizontal or oblique asymptotes of the following rational functions.
a) \begin{align*}y=\frac{3x^2}{x^2+4}\end{align*}
b) \begin{align*}y=\frac{x1}{3x^26}\end{align*}
c) \begin{align*}y=\frac{x^4+1}{x5}\end{align*}
d) \begin{align*}y=\frac{x^33x^2+4x1}{x^22}\end{align*}
Solution
a) When we simplify the function, we get \begin{align*}y=3\frac{12}{x^2+4}\end{align*}
b) We cannot divide the two polynomials. There is a horizontal asymptote at \begin{align*}y = 0\end{align*}
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 \begin{align*}y=x3+\frac{6x7}{x^22}\end{align*}
Practice
Find all asymptotes of the following rational functions:

\begin{align*}\frac{x^2}{x2}\end{align*}
x2x−2 
\begin{align*}\frac{1}{x+4}\end{align*}
1x+4 
\begin{align*}\frac{x^21}{x^2+1}\end{align*}
x2−1x2+1 
\begin{align*}\frac{x4}{x^29}\end{align*}
x−4x2−9 
\begin{align*}\frac{x^2+2x+1}{4x1}\end{align*}
x2+2x+14x−1 
\begin{align*}\frac{x^3+1}{4x1}\end{align*}
x3+14x−1 
\begin{align*}\frac{xx^3}{x^26x7}\end{align*}
x−x3x2−6x−7  \begin{align*}\frac{x^42x}{8x+24}\end{align*}
Graph the following rational functions. Indicate all asymptotes on the graph:
 \begin{align*}\frac{x^2}{x+2}\end{align*}
 \begin{align*}\frac{x^31}{x^24}\end{align*}
 \begin{align*}\frac{x^2+1}{2x4}\end{align*}
 \begin{align*}\frac{xx^2}{3x+2}\end{align*}
Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .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 , where is the voltage or the potential difference, is the current, and is the resistance of the conductor.Image Attributions
Description
Learning Objectives
Here you'll learn how to use division to determine the asymptotes of rational functions.