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Chapter 2 | Limits
lim x →2 f ( x ) =4. From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit . We can think of the limit of a function at a number a as being the one real number L that the functional values approach as the x -values approach a, provided such a real number L exists. Stated more carefully, we have the following definition: Definition Let f ( x ) be a function defined at all values in an open interval containing a , with the possible exception of a itself, and let L be a real number. If all values of the function f ( x ) approach the real number L as the values of x ( ≠ a ) approach the number a , then we say that the limit of f ( x ) as x approaches a is L . (More succinct, as x gets closer to a , f ( x ) gets closer and stays close to L .) Symbolically, we express this idea as (2.3) lim x → a f ( x ) = L .
We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.
Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values 1. To evaluate lim x → a f ( x ), we begin by completing a table of functional values. We should choose two sets of x -values—one set of values approaching a and less than a , and another set of values approaching a and greater than a . Table 2.1 demonstrates what your tables might look like. x f ( x ) x f ( x )
f ( a −0.1)
f ( a +0.1)
a +0.1
a −0.1
f ( a −0.01)
f ( a +0.01)
a +0.01
a −0.01
f ( a −0.001)
f ( a +0.001)
a +0.001
a −0.001
f ( a −0.0001)
f ( a +0.0001)
a +0.0001
a −0.0001
Use additional values as necessary.
Use additional values as necessary.
Table 2.1 Table of Functional Values for lim x → a f ( x )
2. Next, let’s look at the values in each of the f ( x ) columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence f ( a −0.1), f ( a −0.01), f ( a −0.001)., f ( a −0.0001), and so on, and f ( a +0.1), f ( a +0.01), f ( a +0.001), f ( a +0.0001), and so on. ( Note : Although we have chosen the x -values a ±0.1, a ±0.01, a ±0.001, a ±0.0001, and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.) 3. If both columns approach a common y -value L , we state lim x → a f ( x ) = L . We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.
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