Chapter 2 | Limits
143
“does not exist” as DNE. Thus, we would write lim x →0 sin(1/ x ) DNE.) The graph of f ( x ) = sin(1/ x ) is shown in Figure 2.17 and it gives a clearer picture of the behavior of sin(1/ x ) as x approaches 0. You can see that sin(1/ x ) oscillates ever more wildly between −1 and 1 as x approaches 0.
Figure 2.17 The graph of f ( x ) = sin(1/ x ) oscillates rapidly between −1 and 1 as x approaches 0.
| x 2 −4 |
2.6
Use a table of functional values to evaluate lim x →2
if possible.
x −2 ,
One-Sided Limits Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function g ( x ) = | x −2|/( x −2) introduced at the beginning of the section (see Figure 2.12 (b)). As we pick values of x close to2, g ( x ) does not approach a single value, so the limit as x approaches 2 does not exist—that is, lim x →2 g ( x ) DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the x -value 2. To provide a more accurate description, we introduce the idea of a one-sided limit . For all values to the left of 2 (or the negative side of 2), g ( x ) =−1. Thus, as x approaches 2 from the left, g ( x ) approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as lim x →2 − g ( x ) =−1. Similarly, as x approaches 2 from the right (or from the positive side ), g ( x ) approaches 1. Symbolically, we express this idea as lim x →2 + g ( x ) =1. We can now present an informal definition of one-sided limits.
Definition We define two types of one-sided limits .
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