College Algebra 9th txtbk

SECTION 4–4 Rational Functions and Inequalities 303
For example,
f (x) x2 1.44
x 3 x
x2 1.44
x(x 1)(x 1)
has three vertical asymptotes, x 1, x 0, and x 1 (see Fig. 2 on p. 300).
The left and right behavior of some, but not all, rational functions can be described using
the concept of horizontal asymptote. Consider f (x) 1 x. As values of x get larger and larger—
that is, as x increases without bound—the points (x, 1 x) have y coordinates that are positive
and approach 0, as confirmed by Table 3. Similarly, as values of x get smaller and smaller—
that is, as x decreases without bound—the points (x, 1 x) have y coordinates that are negative
and approach 0, as confirmed by Table 4. We write these facts symbolically as
1
as x S and as x S
x S 0
and say that the line y 0 (the x axis) is a horizontal asymptote for the graph of f.
Table 3 Behavior of 1/x as x S
x 1 10 100 1,000 10,000 100,000 1,000,000 . . . x increases without bound (x S )
1x 1 0.1 0.01 0.001 0.0001 0.000 01 0.000 001 . . . 1x approaches 0 (1x S 0)
Table 4 Behavior of 1/x as x S
x 1 10 100 1,000 10,000 100,000 1,000,000 . . . x decreases without bound (x S )
1x 1 0.1 0.01 0.001 0.0001 0.000 01 0.000 001 . . . 1x approaches 0 (1x S 0)
ZZZ EXPLORE-DISCUSS 2
Construct tables similar to Tables 3 and 4 for each of the following functions, and
discuss the behavior of each as x S and as x S :
(A) f (x)
3x (B) g(x)
3x2 (C) h(x) 3x3
x 2 1
x 2 1
x 2 1
Z DEFINITION 3 Horizontal Asymptote
The horizontal line y b is a horizontal asymptote for the graph of y f (x) if
f (x) S b
as
x S
or as
x S
(that is, if f (x) approaches b as x increases without bound or as x decreases without
bound).
A rational function f (x) p(x)q(x) has the same left and right behavior as the quotient of
the leading terms of p(x) and q(x) (Property 5 of Theorem 1). Consequently, a rational function
has at most one horizontal asymptote. Moreover, we can determine easily whether a
rational function has a horizontal asymptote, and if it does, find its equation. Theorem 3
gives the details.