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Horizontal and Vertical Asymptotes for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.

A line y = b is called a horizontal asymptote for the graph of a function f if either

A line x = a is called a vertical asymptote for the graph of a function f if either

Example 1

Find the horizontal and vertical asymptotes of the function

To find the horizontal asymptotes, examine the

Thus, y = 3 is a horizontal asymptote.

To find the vertical asymptotes, look for x values such that the denominator (x – 2) would be 0, in this case, x = 2. Then examine:

  1. the limit of the numerator is 11 and the limit of the denominator is 0 through positive values, and thus,
  2. the limit of the numerator is 11 and the limit of the denominator is 0 through negative values, and thus,

Therefore, x = 2 is a vertical asymptote.

Example 2

Using your calculator, find the horizontal and vertical asymptotes of the function

Enter The graphs shows that as x → ± ∞, the function approaches 0, thus Therefore, a horizontal asymptote is y = 0 (or the x-axis).

For vertical asymptotes, you notice that and Thus, the vertical asymptotes are x = – 2 and x = 2. (See Figure 5.2-7.)

Horizontal and Vertical Asymptotes

Example 3

Using your calculator, find the horizontal and vertical asymptotes of the function

Enter The graph of f (x) shows that as x increases in the first quadrant, f (x) goes higher and higher without bound. As x moves to the left in the 2nd quadrant, f (x) again goes higher and higher without bound. Thus, you may conclude that and and thus, f (x) has no horizontal asymptote. For vertical asymptotes, you notice that Therefore, the line x =0 (or the y -axis) is a vertical asymptote. (See Figure 5.2-8.)

Horizontal and Vertical Asymptotes

Relationship between the limits of rational functions as x → ∞ and horizontal asymptotes: Given then

  1. If the degree of p(x) is same as the degree of q(x), then where a is the coefficient of the highest power of x in p(x) and b is the coefficient of the highest power of x in q(x). The line is a horizontal asymptote. See Example 1 on page 63.
  2. If the degree of p(x) is smaller than the degree of q(x), then The line y = 0 (or x-axis) is a horizontal asymptote. See Example 2 on page 64.
  3. If the degree of p(x) is greater than the degree of q(x), then Thus, f (x) has no horizontal asymptote. See Example 3 on page 64.

Example 4

Using your calculator, find the horizontal asymptotes of the function

Enter The graph shows that f (x) oscillates back and forth about the x-axis. As x → ± ∞, the graph gets closer and closer to the x-axis which implies that f (x) approaches 0. Thus, the line y = 0 (or the x-axis) is a horizontal asymptote. (See Figure 5.2-9.)

Horizontal and Vertical Asymptotes

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.

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