412
Chapter 4 | Applications of Derivatives
Figure 4.46 This function has two horizontal asymptotes.
⎛ ⎝ 3+ 4 x
⎞ ⎠ and lim x
⎛ ⎝ 3+ 4 x
⎞ ⎠ . Determine the horizontal asymptotes of f ( x ) =3+ 4 x , if
4.20
Evaluate lim x
→−∞
→∞
any.
Infinite Limits at Infinity Sometimes the values of a function f become arbitrarily large as x →∞ (or as x →−∞). In this case, we write lim x →∞ f ( x ) =∞ (or lim x →−∞ f ( x ) =∞). On the other hand, if the values of f are negative but become arbitrarily large in magnitude as x →∞ (or as x →−∞), we write lim x →∞ f ( x ) =−∞ (or lim x →−∞ f ( x ) =−∞). For example, consider the function f ( x ) = x 3 . As seen in Table 4.3 and Figure 4.47 , as x →∞ the values f ( x ) become arbitrarily large. Therefore, lim x →∞ x 3 =∞. On the other hand, as x →−∞, the values of f ( x ) = x 3 are negative but become arbitrarily large in magnitude. Consequently, lim x →−∞ x 3 =−∞.
10
20
50
100
1000
x
1000
8000
125,000
1,000,000
1,000,000,000
x 3
−10
−20
−50
−100
−1000
x
−1000 −8000 −125,000 −1,000,000 −1,000,000,000
x 3
Table 4.3 Values of a power function as x →±∞
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