Chapter 4 | Applications of Derivatives
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Figure 4.57 This function has two horizontal asymptotes and it crosses one of the asymptotes.
4.25
3 x 2 +4 x +6
Evaluate lim x →∞
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Determining End Behavior for Transcendental Functions The six basic trigonometric functions are periodic and do not approach a finite limit as x →±∞. For example, sin x oscillates between 1and−1 ( Figure 4.58 ). The tangent function x has an infinite number of vertical asymptotes as x →±∞; therefore, it does not approach a finite limit nor does it approach ±∞ as x →±∞ as shown in Figure 4.59 .
Figure 4.58 The function f ( x ) = sin x oscillates between 1and−1 as x →±∞
Figure 4.59 The function f ( x ) = tan x does not approach a limit and does not approach ±∞ as x →±∞
Recall that for any base b >0, b ≠1, the function y = b x is an exponential function with domain (−∞, ∞) and range (0, ∞). If b >1, y = b x is increasing over ` (−∞, ∞). If 0< b <1, y = b x is decreasing over (−∞, ∞). For the natural exponential function f ( x ) = e x , e ≈2.718>1. Therefore, f ( x ) = e x is increasing on ` (−∞, ∞) and the
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