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Chapter 4 | Applications of Derivatives
Let’s consider several classes of functions here and look at the different types of end behaviors for these functions. End Behavior for Polynomial Functions Consider the power function f ( x ) = x n where n is a positive integer. From Figure 4.50 and Figure 4.51 , we see that lim x →∞ x n =∞; n =1, 2, 3,… and lim x →−∞ x n = ⎨ ∞; n =2, 4, 6,… −∞; n =1, 3, 5,… . ⎧ ⎩
Figure 4.50 For power functions with an even power of n , lim x →∞ x n =∞= lim x →−∞ x n .
Figure 4.51 For power functions with an odd power of n , lim x →∞ x n =∞ and lim x →−∞ x n =−∞.
cx n and lim
cx n , where c is any constant and n is a positive
Using these facts, it is not difficult to evaluate lim x →∞
x →−∞
integer. If c >0, the graph of y = cx n is a vertical stretch or compression of y = x n , and therefore lim x →∞ cx n = lim x →∞ x n and lim x →−∞ cx n = lim x →−∞ x n if c >0. If c <0, the graph of y = cx n is a vertical stretch or compression combined with a reflection about the x -axis, and therefore lim x →∞ cx n =− lim x →∞ x n and lim x →−∞ cx n =− lim x →−∞ x n if c <0.
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