Vertical, Horizontal and Slant Asymptotes of Rational Functions
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Let's think about straight lines for a minute. As you know, most straight lines can be described by the equation \(y = mx + b\). Vertical lines are the exception and we write them as \(x = a\), where \(a\) is a constant.
The three types of asymptotes we talk about on this page cover all possibilities of straight lines. Vertical asymptotes are vertical lines. Horizontal asymptotes are lines with slope zero written as \(y = b\). Slant asymptotes are all other lines, \(y = mx + b\).
We can extract these equations of lines from rational functions. That is what we discuss on this page.
Vertical Asymptotes
As outlined on the basics of rational functions page, vertical asymptotes occur in rational functions at xvalues where the denominator is zero AND the numerator is NOT zero. This implies that even though we have zero in the denominator of the rational function and, therefore, the xvalue is not in the domain, we still need to look at the numerator at that xvalue. If the numerator is not zero then we have a vertical asymptote at that xvalue. (If the numerator IS zero, then there is a hole there. We discuss that on the basics of rational functions page.) Here is a graph and it's corresponding equation showing an example of a vertical asymptote. This graph has an asymptote at \(x = 3\).
\(\displaystyle{ f(x)=\frac{1}{x3} }\)
Vertical Asymptote at \(x = 3 \)
In the case of a vertical asymptote, it doesn't matter if the graph goes off to positive or negative infinity. The xvalue where the denominator is zero is called an asymptote in both cases. Also notice in this graph that one side goes off to positive infinity, the side goes off to negative infinity. Sometimes both sides go off in the same direction, positive or negative. For our discussion, the direction and whether or not they are the same does not matter. (You get to consider thoses cases in calculus.) In all cases, they are just called vertical asymptotes.
Horizontal and Slant Asymptotes
Horizontal and slant asymptotes cover the other two cases of \(y = mx + b\). In these cases, we look at what happens to the yvalue as x gets very large or very small.
Before we get too far into this discussion, make sure you remember how to do long division of polynomials (covered on a separate page). It will also help you to review synthetic division, since it is faster than long division. Synthetic division works in fewer cases but, when can be used, it will save time.
These asymptotes occur as x goes to positive infinity or negative infinity, i.e. as x gets very small or very large. To determine these asymptotes, we follow these steps.
1. Use long division or synthetic division to get a polynomial and rational function where the order of the numerator is less than the order of the denominator. 
2. The asymptote is the polynomial that is not part of the remaining rational function. 
Before we go on, let's watch a couple of videos discussing horizontal asymptotes. Many teachers teach these concepts but without giving you the full context that horizontal asymptotes are just one of several types of these asymptotes.
A couple of quick examples will help. In these examples, we have already separated the rational function using long division. So the remaining part that is still a rational function has a smaller order numerator than denominator.
Example 1  \(\displaystyle{f(x)=1+\frac{1}{x}}\)
Example 1   What is the asymptote of \(\displaystyle{f(x)=1+\frac{1}{x}}\)?
As x gets larger and larger, \(1/x\) gets smaller and smaller and eventually will become zero. This leaves only \(y=1\) for very large x. The same argument holds as x moves in the negative direction. So \(y=1\) is the asymptote, a horizontal asymptote in fact.
Example 2  \(\displaystyle{f(x)=x+3+\frac{1}{x}}\)
Example 2   What is the asymptote of \(\displaystyle{f(x)=x+3+\frac{1}{x}}\)?
Using the same logic as in example 1, the asymptote is \(y=x+3\), in this case a slant asymptote.
Similar results hold for other types of problems. The key is to use long division to get the form we need.
Okay, time for the practice problems.
Practice
Unless otherwise instructed, calculate all horizontal and slant asymptotes for these functions. You do not need to sketch a graph.
\(\displaystyle{ f(x) = \frac{1}{x} }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{1}{x} }\).
Solution
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\(\displaystyle{ f(x) = \frac{7}{x3} }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{7}{x3} }\).
Solution
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\(\displaystyle{ f(x) = \frac{5}{x+1} + 3 }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{5}{x+1} + 3 }\).
Solution
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\(\displaystyle{ f(x) = \frac{8x6}{x^2+3x} }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{8x6}{x^2+3x} }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{8x6}{x^2+3x}  9 }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{8x6}{x^2+3x}  9 }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{2x3}{x+4} }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{2x3}{x+4} }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{12x6}{3x+4} + 2 }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{12x6}{3x+4} + 2 }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{58x^2}{2x^2+5}  5 }\)
Problem Statement
Calculate all horizontal and slant asymptotes of the function \(\displaystyle{ f(x) = \frac{58x^2}{2x^2+5}  5 }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{x^2+5x+6}{x+3} }\)
Problem Statement
Calculate all horizontal and slant asymptotes for the function \(\displaystyle{ f(x) = \frac{x^2+5x+6}{x+3} }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{2x^38x+16}{x^2+4} }\)
Problem Statement
Calculate all horizontal and slant asymptotes of the function \(\displaystyle{ f(x) = \frac{2x^38x+16}{x^2+4} }\). You do not need to sketch a graph.
Solution
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\(\displaystyle{ f(x) = \frac{5x^46x+2}{7x3} }\).
Problem Statement
Calculate all horizontal and slant asymptotes of the function \(\displaystyle{ f(x) = \frac{5x^46x+2}{7x3} }\). You do not need to sketch a graph.
Solution
He is correct when he says that a slant asymptote occurs only when the difference between the highest powers is exactly one and the higher power is in the numerator. What he doesn't say is that there is still an asymptote but it is neither horizontal or a straight line. In this case the asymptote will get a cubic function. Use a plotting program to see the graph. It is quite interesting and then see if you can get the equation of the asymptote from the equation. You have all the tools necessary to understand how to do it.
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Find the slant asymptote of \(\displaystyle{f(x)=\frac{x^2+x1}{x1}}\).
Problem Statement
Find the slant asymptote of \(\displaystyle{f(x)=\frac{x^2+x1}{x1}}\).
Solution
video by Krista King Math 

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Find the zeroes and vertical asymptotes of the function \(\displaystyle{f(x)=\frac{6x^2+7x+2}{4x4}}\).
Problem Statement 

Find the zeroes and vertical asymptotes of the function \(\displaystyle{f(x)=\frac{6x^2+7x+2}{4x4}}\).
Hint 

Remember, a zero is where the numerator is zero but the denominator is NOT zero.
Problem Statement 

Find the zeroes and vertical asymptotes of the function \(\displaystyle{f(x)=\frac{6x^2+7x+2}{4x4}}\).
Final Answer 

zeroes: \(x=2/3\) and \(x=1/2\)
VA: \(x=1\)
Problem Statement
Find the zeroes and vertical asymptotes of the function \(\displaystyle{f(x)=\frac{6x^2+7x+2}{4x4}}\).
Hint
Remember, a zero is where the numerator is zero but the denominator is NOT zero.
Solution
video by PatrickJMT 

Final Answer
zeroes: \(x=2/3\) and \(x=1/2\)
VA: \(x=1\)
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Find the slant asymptote of \(\displaystyle{f(x)=\frac{x^3}{x^21}}\)
Problem Statement 

Find the slant asymptote of \(\displaystyle{f(x)=\frac{x^3}{x^21}}\)
Final Answer 

\( y=x \)
Problem Statement
Find the slant asymptote of \(\displaystyle{f(x)=\frac{x^3}{x^21}}\)
Solution
\[\begin{array}{r} \\ x^21 ~ \\ \phantom{.} \\ \phantom{.} \\ \end{array} \begin{array}{rrrr} & & x & \\ \hline x^3 & & & \\ x^3 & & +x & \\ \hline & & +x & \end{array}\]
So first we need to do polynomial long division to get an expression with a rational function where the highest power of the numerator is less than the highest power of the denominator. We did that above to give us this expression. 
\(\displaystyle{ f(x)=\frac{x^3}{x^21} = x + \frac{x}{x^21} }\) 
The slant asymptote is the expression in front of the rational function, i.e. \( g(x) = x \). 
Although not asked for in the problem statement, we plotted \(f(x)\) and the slant asymptote \(g(x)\) on the same set of axes. The plot is shown below. 
Final Answer
\( y=x \)
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Find all the asymptotes of \(\displaystyle{f(x)=\frac{5x^2}{x+3}}\).
Problem Statement
Find all the asymptotes of \(\displaystyle{f(x)=\frac{5x^2}{x+3}}\).
Solution
video by PatrickJMT 

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Find the asymptotes of \(\displaystyle{f(x)=\frac{x^2+4x+7}{x1}}\).
Problem Statement
Find the asymptotes of \(\displaystyle{f(x)=\frac{x^2+4x+7}{x1}}\).
Solution
video by PatrickJMT 

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Determine the intercepts and asymptotes of \(\displaystyle{f(x)=\frac{2x^29x5}{2x^2+5x3}}\).
Problem Statement
Determine the intercepts and asymptotes of \(\displaystyle{f(x)=\frac{2x^29x5}{2x^2+5x3}}\).
Solution
video by MIP4U 

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Determine the intercepts, asymptotes and holes of \(\displaystyle{f(x)=\frac{x^2x12}{2x^24x16}}\).
Problem Statement
Determine the intercepts, asymptotes and holes of \(\displaystyle{f(x)=\frac{x^2x12}{2x^24x16}}\).
Solution
video by MIP4U 

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Find the domain, asymptotes and holes of \(\displaystyle{f(x)=\frac{x+2}{x^2+5x+6}}\).
Problem Statement
Find the domain, asymptotes and holes of \(\displaystyle{f(x)=\frac{x+2}{x^2+5x+6}}\).
Solution
video by Dr Phil Clark 

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Find the asymptotes and zeroes of \(\displaystyle{R(x)=\frac{2(x4)(x+3)}{(x1)(x2)}}\).
Problem Statement
Find the asymptotes and zeroes of \(\displaystyle{R(x)=\frac{2(x4)(x+3)}{(x1)(x2)}}\).
Solution
video by Dr Phil Clark 

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Find the holes and asymptotes of \(\displaystyle{f(x)=\frac{x^2x2}{x^26x+8}}\).
Problem Statement
Find the holes and asymptotes of \(\displaystyle{f(x)=\frac{x^2x2}{x^26x+8}}\).
Solution
video by Dr Phil Clark 

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