Chapter 4 | Applications of Derivatives
465
Growth Rates of Functions Suppose the functions f and g both approach infinity as x →∞. Although the values of both functions become arbitrarily large as the values of x become sufficiently large, sometimes one function is growing more quickly than the other. For example, f ( x ) = x 2 and g ( x ) = x 3 both approach infinity as x →∞. However, as shown in the following table, the values of x 3 are growing much faster than the values of x 2 .
10,000
10
100
1000
x
f ( x ) = x 2
10,000
1,000,000
100,000,000
100
g ( x ) = x 3
1000 1,000,000 1,000,000,000 1,000,000,000,000
Table 4.7 Comparing the Growth Rates of x 2 and x 3
In fact,
x 3 x 2
x 2 x 3
1 x =0.
lim x →∞
= lim x →∞
x = ∞.or, equivalently, lim x →∞
= lim x →∞
As a result, we say x 3 is growing more rapidly than x 2 as x →∞. On the other hand, for f ( x ) = x 2 and g ( x ) =3 x 2 +4 x +1, although the values of g ( x ) are always greater than the values of f ( x ) for x >0, each value of g ( x ) is roughly three times the corresponding value of f ( x ) as x →∞, as shown in the following table. In fact, lim x →∞ x 2 3 x 2 +4 x +1 = 1 3 . x 10 100 1000 10,000
f ( x ) = x 2
100 10,000 1,000,000 100,000,000
g ( x ) =3 x 2 +4 x +1
341 30,401 3,004,001 300,040,001
Table 4.8 Comparing the Growth Rates of x 2 and 3 x 2 +4 x +1
In this case, we say that x 2 and 3 x 2 +4 x +1 are growing at the same rate as x →∞. More generally, suppose f and g are two functions that approach infinity as x →∞. We say g grows more rapidly than f as x →∞ if lim x →∞ g ( x ) f ( x ) = ∞; or, equivalently, lim x →∞ f ( x ) g ( x ) =0. On the other hand, if there exists a constant M ≠0 such that lim x →∞ f ( x ) g ( x ) = M , we say f and g grow at the same rate as x →∞.
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