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:

- the limit of the numerator is 11 and the limit of the denominator is 0 through positive values, and thus,
- 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.)

### 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.)

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

- 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. - 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. - 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.)

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

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