Chapter 4 | Applications of Derivatives
467
5
10
15
20
x
x 2
25
100
225
400
e x
148 22,026 3,269,017 485,165,195
Table 4.9 Growth rates of a power function and an exponential function.
x 2 =∞, we can use L’Hôpital’s rule to evaluate lim x →∞ ln x x 2 . We
b. Since lim x
→∞ ln
x =∞ and lim x →∞
obtain
1/ x 2 x = lim x →∞
ln x x 2
1 2 x 2
lim x →∞
= lim x →∞
=0.
Thus, x 2 grows more rapidly than ln x as x →∞ (see Figure 4.74 and Table 4.10 ).
Figure 4.74 A power function grows at a faster rate than a logarithmic function.
10,000
10
100
1000
x
ln( x )
6.908
9.210
2.303
4.605
x 2
10,000 1,000,000 100,000,000
100
Table 4.10 Growth rates of a power function and a logarithmic function
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