In this section we will explore asymptotes of rational functions. In particular, we
will look at horizontal, vertical, and oblique asymptotes. Keep in mind that we
are studying a rational function of the form,
where P(x) and Q(x) are polynomials. We say that f(x) is in lowest terms if
P(x) and Q(x) have no common factors.
1. Horizontal Asymptotes
A. Degree of P(x) < Degree of Q(x)
The rational function f(x) = P(x)
/ Q(x) in lowest terms has horizontal
asymptote
y = 0 if the degree of the numerator, P(x), is less than
the degree of denominator, Q(x). In this case, f(x) → 0 as x → ±∞.
An example of a function with horizontal asymptote y = 0 is,
B. Degree of P(x) = Degree of Q(x)
The rational function f(x) = P(x) /
Q(x) in lowest terms has horizontal
asymptote
y = a / b if the degree of the numerator, P(x), is equal to
the degree of denominator, Q(x), where a is the leading coefficient of
P(x) and b is leading coefficient of Q(x). In this case, f(x) → a /
b as
x → ±∞. An example of a function with horizontal asymptote y = 1 /
3
is,
C. Degree of P(x) > Degree of Q(x)
The rational function f(x) = P(x) /
Q(x) in lowest terms has no horizontal
asymptotes if the degree of the numerator, P(x), is greater than the
degree of denominator, Q(x).
2. Vertical Asymptotes
The rational function f(x) = P(x) /
Q(x) in lowest terms has vertical asymptotes,
x = x_{1}, x = x_{2}, . . . , x = x_{k}, where x_{1}, x_{2}, . . . , x_{k} are roots of Q(x). To
find the vertical asymptotes of f(x) be sure that it is in lowest terms by canceling
any common factors, and then find the roots of Q(x).
3. Oblique Asymptotes
The rational function f(x) = P(x)
/ Q(x) in lowest terms has an oblique asymptote
if the degree of the numerator, P(x), is exactly one greater than the degree
of the denominator, Q(x). You can find oblique asymptotes using polynomial
division, where the quotient is the equation of the oblique asymptote.
For example, the function,
has oblique asymptote found by polynomial division,
Thus, we found that,
and the equation of the oblique asymptote is the quotient,
y = x + 2.
That is, as x gets large, f(x) approaches the line y = x + 2 as depicted in
the graph below. Notice that the remainder approaches 0 as x → ∞.
Finding Asymptotes
When finding asymptotes, it is important to remember that f(x) = P(x) /
Q(x) must be
in lowest terms. For example, just because x = a is a simple root of Q(x) does
not mean that it corresponds to a vertical asymptote. In particular, if x = a is
a simple root of P(x) and Q(x) the graph will have a hole at the point x = a.
Since a point has no size, a graph of such a function will not look any different
when graphed on a calculator. Therefore, we denote a hole in a graph with an
open circle. For example, consider the function,
Notice that we can write Q(x) = x^{2}− 1 as Q(x) = (x − 1)(x + 1). Notice
that x = 1 and
x = −1 are not in the domain of f(x) since they are zeros of
Q(x). However, if x ≠ 1 we can cancel the factor x − 1 in the numerator and
denominator as,
Notice that f(x) and agree everywhere except at the point x = 1
(since x = 1 is not in the domain of f(x), but is in the domain of g(x)). Therefore,
the graph of f(x) looks identical to the graph of g(x) except that there is a hole
in the graph of f(x) at x = 1 as depicted below.
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In the next section we will explore the graphs of rational functions.
Graphing Rational Functions
