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Chapter 4 | Applications of Derivatives
horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as f ( x ) = p ( x ) q ( x ) = g ( x )+ r ( x ) q ( x ) ,
where the degree of r ( x ) is less than the degree of q ( x ). As a result, lim x →±∞
r ( x )/ q ( x ) =0. Therefore, the values
of ⎡
⎣ f ( x )− g ( x ) ⎤
⎦ approach zero as x →±∞. If the degree of p ( x ) is exactly one more than the degree of q ( x )
( n = m +1), the function g ( x ) is a linear function. In this case, we call g ( x ) an oblique asymptote. Now let’s consider the end behavior for functions involving a radical. Example 4.26 Determining End Behavior for a Function Involving a Radical Find the limits as x →∞ and x →−∞ for f ( x ) = 3 x −2 4 x 2 +5 and describe the end behavior of f . Solution Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of x . To determine the appropriate power of x , consider the expression 4 x 2 +5 in the denominator. Since 4 x 2 +5≈ 4 x 2 =2| x | for large values of x in effect x appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by | x |. Then, using the fact that | x | = x for x >0, | x | =− x for x <0, and | x | = x 2 for all x , we calculate the limits as follows: lim x →∞ 3 x −2 4 x 2 +5 = lim x →∞ (1/| x |)(3 x −2) (1/| x |) 4 x 2 +5
(1/ x )(3 x −2) ⎛ ⎝ 1/ x 2 ⎞ ⎠ ⎛ ⎝ 4 x 2 +5
= lim x →∞
⎞ ⎠
3−2/ x 4+5/ x 2
= lim x →∞
= 3
= 3 2
4
(1/| x |)(3 x −2) (1/| x |) 4 x 2 +5 (−1/ x )(3 x −2) ⎛ ⎝ 1/ x 2 ⎛ ⎝ 4 x 2 +5 ⎞ ⎠
3 x −2 4 x 2 +5
lim x →−∞
= lim x →−∞
= lim x →−∞
⎞ ⎠
−3+2/ x 4+5/ x 2
= lim x →−∞
= −3 4
= −3 2 .
Therefore, f ( x ) approaches the horizontal asymptote y = 3 2 as x →−∞ as shown in the following graph.
as x →∞ and the horizontal asymptote y = − 3 2
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