Chapter 2 | Limits
137
4. Using a graphing calculator or computer software that allows us graph functions, we can plot the function f ( x ), making sure the functional values of f ( x ) for x -values near a are in our window. We can use the trace feature to move along the graph of the function and watch the y -value readout as the x -values approach a . If the y -values approach L as our x -values approach a from both directions, then lim x → a f ( x ) = L . Wemay need to zoom in on our graph and repeat this process several times.
We apply this Problem-Solving Strategy to compute a limit in Example 2.4 . Example 2.4 Evaluating a Limit Using a Table of Functional Values 1
sin x x using a table of functional values.
Evaluate lim x →0
Solution We have calculated the values of f ( x ) = (sin x )/ x for the values of x listed in Table 2.2 .
sin x x
sin x x
x
x
−0.1
0.998334166468
0.1
0.998334166468
−0.01
0.999983333417
0.01
0.999983333417
−0.001
0.999999833333
0.001
0.999999833333
−0.0001 0.999999998333
0.0001 0.999999998333
Table 2.2 Table of Functional Values for lim x →0
sin x x
Note : The values in this table were obtained using a calculator and using all the places given in the calculator output. As we read down each (sin x ) x column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that lim x →0 sin x x =1. A calculator or computer-generated graph of f ( x ) = (sin x ) x would be similar to that shown in Figure 2.13 , and it confirms our estimate.
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