418
Chapter 4 | Applications of Derivatives
Figure 4.52 The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.
10
100
1000
x
f ( x ) =5 x 3 −3 x 2 +4
4,970,004
4,997,000,004
4704
g ( x ) =5 x 3
5,000,000
5,000,000,000
5000
−10
−100
−1000
x
f ( x ) =5 x 3 −3 x 2 +4
−5296 −5,029,996 −5,002,999,996
g ( x ) =5 x 3
−5000 −5,000,000 −5,000,000,000
Table 4.4 A polynomial’s end behavior is determined by the term with the largest exponent.
End Behavior for Algebraic Functions The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In Example 4.25 , we show that the limits at infinity of a rational function f ( x ) = depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of x appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of x . Example 4.25 Determining End Behavior for Rational Functions For each of the following functions, determine the limits as x →∞ and x →−∞. Then, use this information to describe the end behavior of the function. a. f ( x ) = 3 x −1 2 x +5 ( Note: The degree of the numerator and the denominator are the same.) p ( x ) q ( x )
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