## 17Calculus Precalculus - Asymptotes of Rational Functions

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Vertical, Horizontal and Slant Asymptotes of Rational Functions

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 x-values 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 x-value is not in the domain, we still need to look at the numerator at that x-value. If the numerator is not zero then we have a vertical asymptote at that x-value. (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 $$x = 3$$.

$$\displaystyle{ f(x)=\frac{1}{x-3} }$$
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 x-value 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 y-value 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. So keep that in mind as you watch these videos. She shows some great examples here too.

### Krista King Math - Horizontal Asymptotes - Basic Overview [4min-42secs]

video by Krista King Math

### Krista King Math - Horizontal Asymptotes - Further Detail [11min-4secs]

video by Krista King Math

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 the function $$\displaystyle{ f(x) = \frac{1}{x} }$$.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{1}{x} }$$.

Solution

### 2950 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{7}{x-3} }$$.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{7}{x-3} }$$.

Solution

### 2951 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5}{x+1} + 3 }$$.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5}{x+1} + 3 }$$.

Solution

### 2952 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{8x-6}{x^2+3x} }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{8x-6}{x^2+3x} }$$. You do not need to sketch a graph.

Solution

### 2953 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{8x-6}{x^2+3x} - 9 }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{8x-6}{x^2+3x} - 9 }$$. You do not need to sketch a graph.

Solution

### 2954 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{2x-3}{x+4} }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{2x-3}{x+4} }$$. You do not need to sketch a graph.

Solution

### 2955 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{12x-6}{3x+4} + 2 }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{12x-6}{3x+4} + 2 }$$. You do not need to sketch a graph.

Solution

### 2956 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5-8x^2}{2x^2+5} - 5 }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5-8x^2}{2x^2+5} - 5 }$$. You do not need to sketch a graph.

Solution

### 2957 video

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Unless otherwise instructed, 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.

Problem Statement

Unless otherwise instructed, 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

### 2958 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{2x^3-8x+16}{x^2+4} }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{2x^3-8x+16}{x^2+4} }$$. You do not need to sketch a graph.

Solution

### 2959 video

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Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5x^4-6x+2}{7x-3} }$$. You do not need to sketch a graph.

Problem Statement

Unless otherwise instructed, calculate all horizontal and slant asymptotes for the function $$\displaystyle{ f(x) = \frac{5x^4-6x+2}{7x-3} }$$. 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.

### 2960 video

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Find the slant asymptote of $$\displaystyle{f(x)=\frac{x^2+x-1}{x-1}}$$.

Problem Statement

Find the slant asymptote of $$\displaystyle{f(x)=\frac{x^2+x-1}{x-1}}$$.

Solution

### 1685 video

video by Krista King Math

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

Problem Statement

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

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 $$\displaystyle{f(x)=\frac{6x^2+7x+2}{4x-4}}$$.

zeroes: $$x=-2/3$$ and $$x=-1/2$$
VA: $$x=1$$

Problem Statement

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

Hint

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

Solution

### 2041 video

video by PatrickJMT

zeroes: $$x=-2/3$$ and $$x=-1/2$$
VA: $$x=1$$

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Find the slant asymptote of $$\displaystyle{f(x)=\frac{x^3}{x^2-1}}$$.

Problem Statement

Find the slant asymptote of $$\displaystyle{f(x)=\frac{x^3}{x^2-1}}$$.

Solution

### 1666 video

video by Krista King Math

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Find all the asymptotes of $$\displaystyle{f(x)=\frac{5-x^2}{x+3}}$$.

Problem Statement

Find all the asymptotes of $$\displaystyle{f(x)=\frac{5-x^2}{x+3}}$$.

Solution

### 1671 video

video by PatrickJMT

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Find the asymptotes of $$\displaystyle{f(x)=\frac{x^2+4x+7}{x-1}}$$.

Problem Statement

Find the asymptotes of $$\displaystyle{f(x)=\frac{x^2+4x+7}{x-1}}$$.

Solution

### 1673 video

video by PatrickJMT

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Determine the intercepts and asymptotes of $$\displaystyle{f(x)=\frac{2x^2-9x-5}{2x^2+5x-3}}$$.

Problem Statement

Determine the intercepts and asymptotes of $$\displaystyle{f(x)=\frac{2x^2-9x-5}{2x^2+5x-3}}$$.

Solution

### 1674 video

video by MIP4U

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Determine the intercepts, asymptotes and holes of $$\displaystyle{f(x)=\frac{x^2-x-12}{2x^2-4x-16}}$$.

Problem Statement

Determine the intercepts, asymptotes and holes of $$\displaystyle{f(x)=\frac{x^2-x-12}{2x^2-4x-16}}$$.

Solution

### 1677 video

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

### 1682 video

video by Dr Phil Clark

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Find the asymptotes and zeroes of $$\displaystyle{R(x)=\frac{-2(x-4)(x+3)}{(x-1)(x-2)}}$$.

Problem Statement

Find the asymptotes and zeroes of $$\displaystyle{R(x)=\frac{-2(x-4)(x+3)}{(x-1)(x-2)}}$$.

Solution

### 1683 video

video by Dr Phil Clark

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Find the holes and asymptotes of $$\displaystyle{f(x)=\frac{x^2-x-2}{x^2-6x+8}}$$.

Problem Statement

Find the holes and asymptotes of $$\displaystyle{f(x)=\frac{x^2-x-2}{x^2-6x+8}}$$.

Solution

### 1684 video

video by Dr Phil Clark

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Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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