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 . Notice the function approaches negative infinity as x approaches 0 from the left and that it approaches positive infinity as x approaches 0 from the right. 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 has a horizontal asymptote y = 0, as shown below. Comparing Vertical and Horizontal Asymptotes A rational function is undefined at a vertical asymptote. The limits as or as will be the same if the function has a horizontal asymptote. 7.1.1 Graph the function in a [20, 20, 5] x [10, 10, 2] window. Use the graph to estimate the vertical and horizontal asymptotes and write their equations. Click here for the answer.
Vertical Asymptotes from a Table of Values A table of values of the function near the vertical asymptote x = 3 can be useful in illustrating the function's behavior near the asymptote.
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, can be written as Because is the same as except when x = 2, the expression can be used in left and righthand limits to determine the behavior of the function near x = 3. The LeftHand Limit As x approaches 3 from the left the numerator of approaches 6 while the denominator approaches 0 through negative values. This means the fraction will approach negative infinity. The RightHand Limit As x approaches 3 from the right the numerator of approaches 6 while the denominator approaches 0 through positive values. This means the fraction will approach positive infinity. 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 as it approaches the horizontal asymptote y = 2 when x is large and positive and when x is negative and large in absolute value.
7.1.3 What do the two tables tell you about the behavior of the function as gets large? Click here for the answer. Finding a Horizontal Asymptote Analytically
As the magnitude of x gets large, the term 2x^{2} dominates the numerator and the term x^{2} dominates the denominator. The end behavior of the rational function
resembles the rational function
. We say that y = 2 is an end behavior model for
. In other words, the line 

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