Module 7 - Limits and Infinity | ||||||||||||||||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self Test | ||||||||||||||||||||||||
Lesson 7.1: Vertical and Horizontal Asymptotes | ||||||||||||||||||||||||
In this lesson you will investigate a
Conceptually, an asymptote is a line or a curve that the graph of a function approaches. Vertical asymptotes occur where function value magnitudes grow larger as x approaches a fixed number. Horizontal asymptotes occur when a function approaches a horizontal line as x approaches positive or negative infinity. Both types of asymptotes are discussed in this lesson and their formal definitions are given below. Vertical Asymptote Definition A function has a vertical asymptote at x = a if as the input values approach a from at least one side, the magnitude of the output increases without bound. That is, the line x = a is a vertical asymptote if ![]()
The diagram below illustrates the vertical asymptote of the function
![]() ![]() Horizontal Asymptote Definition A horizontal asymptote is a horizontal line that the graph of a function approaches as the magnitude of the input increases without bound in either a positive or negative direction. That is, the line y = k is a horizontal asymptote of f(x) if ![]() A function may cross a horizontal asymptote for finite values of the input.
The function
![]() ![]() Comparing Vertical and Horizontal Asymptotes
A rational function is undefined at a vertical asymptote. The limits as
7.1.1 Graph the function
Vertical Asymptotes from a Table of Values
A table of values of the function
![]() 7.1.2 Describe the behavior of the function near the vertical asymptote x = 3. Click here for the answer. Finding a Vertical Asymptote Analytically
Factoring the numerator and denominator of a rational function and simplifying the fraction can provide information about the function near a possible asymptote. For example,
![]()
Because
The Left-Hand Limit
As x approaches 3 from the left the numerator of
![]() The Right-Hand Limit
As x approaches 3 from the right the numerator of
![]()
Therefore, x = 3 is a vertical asymptote of
Horizontal Asymptotes from a Table of Values
Make a table of values to show the behavior of the function
7.1.3 What do the two tables tell you about the behavior of the function
Finding a Horizontal Asymptote Analytically
As the magnitude of x gets large, the term 2x2 dominates the numerator and the term x2 dominates the denominator. The end behavior of the rational function
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