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Copyright © Cengage Learning. All rights reserved Introduction to Limits

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2 What You Should Learn Understand the limit concept. Use the definition of a limit to estimate limits. Determine whether limits of functions exist. Use properties of limits and direct substitution to evaluate limits.

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3 Definition of Limit

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5 Example 2 – Estimating a Limit Numerically Use a table to estimate the limit numerically Solution: Let f (x) = 3x – 2. Then construct a table that shows values of f (x) for two sets of x-values—one set that approaches 2 from the left and one that approaches 2 from the right.

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6 Example 2 – Solution From the table, it appears that the closer x gets to 2, the closer f (x) gets to 4. So, you can estimate the limit to be 4. Figure 11.2 adds further support to this conclusion. Figure 11.2 cont’d

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7 Limits That Fail to Exist

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8 Example 6 – Comparing Left and Right Behavior Show that the limit does not exist. Solution: Consider the graph of the function given by f (x) = | x | /x. In Figure 11.7, you can see that for positive x-values and for negative x-values Figure 11.7

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9 Example 6 – Solution This means that no matter how close x gets to 0, there will be both positive and negative x-values that yield f (x) = 1 and f (x) = –1. This implies that the limit does not exist. cont’d

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10 Limits That Fail to Exist

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11 Properties of Limits and Direct Substitution

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12 Properties of Limits and Direct Substitution You have seen that sometimes the limit of f (x) as x → c is simply f (c). In such cases, it is said that the limit can be evaluated by direct substitution. That is, There are many “well-behaved” functions, such as polynomial functions and rational functions with nonzero denominators, that have this property. Substitute c for x.

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13 Properties of Limits and Direct Substitution Some of the basic ones are included in the following list.

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14 Properties of Limits and Direct Substitution

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15 Example 9 – Direct Substitution and Properties of Limits Direct Substitution Scalar Multiple Property Quotient Property Product Property Direct Substitution

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16 Example 9 – Direct Substitution and Properties of Limits Sum and Power Properties cont’d

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17 Properties of Limits and Direct Substitution The results of using direct substitution to evaluate limits of polynomial and rational functions are summarized as follows.

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