Chapter 4 | Applications of Derivatives
417
If c =0, y = cx n =0, in which case lim x →∞
cx n =0= lim
cx n .
x →−∞
Example 4.24 Limits at Infinity for Power Functions
f ( x ) and lim x
f ( x ).
For each function f , evaluate lim x →∞
→−∞
a. f ( x ) =−5 x 3 b. f ( x ) =2 x 4
Solution a. Since the coefficient of x 3 is −5, the graph of f ( x ) =−5 x 3 involves a vertical stretch and reflection of the graph of y = x 3 about the x -axis. Therefore, lim x →∞ ⎛ ⎝ −5 x 3 ⎞ ⎠ =−∞ and lim x →−∞ ⎛ ⎝ −5 x 3 ⎞ ⎠ =∞. b. Since the coefficient of x 4 is 2, the graph of f ( x ) =2 x 4 is a vertical stretch of the graph of y = x 4 . Therefore, lim x →∞ 2 x 4 =∞ and lim x →−∞ 2 x 4 =∞.
Let f ( x ) =−3 x 4 . Find lim x →∞
4.23
f ( x ).
f ( x ) for any polynomial
We now look at how the limits at infinity for power functions can be used to determine lim x →±∞
function f . Consider a polynomial function
n −1 +…+ a
n + a
f ( x ) = a n x
n −1 x
1 x + a 0
of degree n ≥1 so that a n ≠0. Factoring, we see that
f ( x ) = a n x ⎞ ⎠ . As x →±∞, all the terms inside the parentheses approach zero except the first term. We conclude that lim x →±∞ f ( x ) = lim x →±∞ a n x n . For example, the function f ( x ) =5 x 3 −3 x 2 +4 behaves like g ( x ) =5 x 3 as x →±∞ as shown in Figure 4.52 and Table 4.4 . n ⎛ ⎝ 1+ a n −1 a n 1 x +…+ a 1 a n 1 x n −1 + a 0 a n 1 x n
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