424
Chapter 4 | Applications of Derivatives
x approaches ∞ as x →∞ and approaches 0 as x →−∞ as
range is ` (0, ∞). The exponential function f ( x ) = e shown in Table 4.5 and Figure 4.60 .
−5
0
5
−2
2
x
e x
0.00674 0.135 1 7.389 148.413
Table 4.5 End behavior of the natural exponential function
Figure 4.60 The exponential function approaches zero as x →−∞ and approaches ∞ as x →∞.
Recall that the natural logarithm function f ( x ) = ln( x ) is the inverse of the natural exponential function y = e x . Therefore, the domain of f ( x ) = ln( x ) is (0, ∞) and the range is (−∞, ∞). The graph of f ( x ) = ln( x ) is the reflection of the graph of y = e x about the line y = x . Therefore, ln( x )→−∞ as x →0 + and ln( x )→∞ as x →∞ as shown in Figure 4.61 and Table 4.6 .
0.01
0.1
10
100
1
x
ln( x )
−4.605 −2.303 0 2.303 4.605
Table 4.6 End behavior of the natural logarithm function
Figure 4.61 The natural logarithm function approaches ∞ as x →∞.
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