Chapter 4 | Applications of Derivatives
421
3 x 2 +2 x −1 5 x 2 −4 x +7
4.24
lim x →±∞
Evaluate
and use these limits to determine the end behavior of
2 +2 x −1 5 x 2 −4 x +7 .
f ( x ) = 3 x
Before proceeding, consider the graph of f ( x ) = ⎛ shown in Figure 4.56 . As x →∞ and x →−∞, the graph of f appears almost linear. Although f is certainly not a linear function, we now investigate why the graph of f seems to be approaching a linear function. First, using long division of polynomials, we can write f ( x ) = 3 x 2 +4 x x +2 =3 x −2+ 4 x +2 . Since 4 ( x +2) →0 as x →±∞, we conclude that lim x →±∞ ⎛ ⎝ f ( x )−(3 x −2) ⎞ ⎠ = lim x →±∞ 4 x +2 =0. Therefore, the graph of f approaches the line y =3 x −2 as x →±∞. This line is known as an oblique asymptote for f ( Figure 4.56 ). ⎝ 3 x 2 +4 x ⎞ ⎠ ( x +2)
Figure 4.56 The graph of the rational function f ( x ) = ⎛ ⎝ 3 x 2 +4 x ⎞
⎠ /( x +2) approaches the oblique asymptote
y =3 x −2as x →±∞.
We can summarize the results of Example 4.25 to make the following conclusion regarding end behavior for rational functions. Consider a rational function
n −1 +…+ a
n + a m + b
a n x
n −1 x
1 x + a 0
p ( x ) q ( x ) =
f ( x ) =
,
m −1 +…+ b
b m x
m −1 x
1 x + b 0
where a n ≠0and b m ≠0. 1. If the degree of the numerator is the same as the degree of the denominator ( n = m ), then f has a horizontal asymptote of y = a n / b m as x →±∞. 2. If the degree of the numerator is less than the degree of the denominator ( n < m ), then f has a horizontal asymptote of y =0 as x →±∞. 3. If the degree of the numerator is greater than the degree of the denominator ( n > m ), then f does not have a
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