Chapter 2 | Limits
147
Example 2.9 Recognizing an Infinite Limit
Evaluate each of the following limits, if possible. Use a table of functional values and graph f ( x ) =1/ x to confirm your conclusion. a. lim x →0 − 1 x b. lim x →0 + 1 x c. lim x →0 1 x
Solution Begin by constructing a table of functional values.
1 x
1 x
x
x
−0.1
−10
0.1
10
−0.01
−100
0.01
100
−0.001
−1000
0.001
1000
−0.0001
−10,000
0.0001
10,000
−0.00001
−100,000
0.00001
100,000
−0.000001 −1,000,000
0.000001 1,000,000
Table 2.7 Table of Functional Values for f ( x ) = 1 x a. The values of 1/ x decrease without bound as x approaches 0 from the left. We conclude that lim x →0 − 1 x =−∞. b. The values of 1/ x increase without bound as x approaches 0 from the right. We conclude that lim x →0 + 1 x =+∞. c. Since lim x →0 − 1 x =−∞ and lim x →0 + 1 x =+∞ have different values, we conclude that lim x →0 1 x DNE. The graph of f ( x ) =1/ x in Figure 2.19 confirms these conclusions.
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