SECTION 14.1 Investigating Limits Using Tables and Graphs 941 14.1 Investigating Limits Using Tables and Graphs Now Work the ‘Are You Prepared?’ problems on page 944. • Piecewise–defined Functions (Section 2.4, pp. 105–107) PREPARING FOR THIS SECTION Before getting started, review the following: Need to Review? Interval notation is discussed in Section A.9, pp. A76 – A78 . For all x approximately equal to the number c , with ≠ x c, the corresponding value of f is approximately equal to the number N . As x gets closer to c, but remains unequal to c , the corresponding value of f gets closer to N . OBJECTIVES 1 Investigate a Limit Using a Table (p. 941) 2 Investigate a Limit Using a Graph (p. 943) The Idea of a Limit The idea of a limit of a function is what connects algebra and geometry to the mathematics of calculus. In working with the limit of a function, we encounter notation of the form ( ) = → f x N lim x c This is read as “the limit of ( ) f x as x approaches c equals the number N .” Here f is a function defined on some open interval containing the number c . However, f need not be defined at c . The meaning of ( ) = → f x N lim x c may be described as follows: Another description of ( ) = → f x N lim x c is 1 Investigate a Limit Using a Table Tables generated with the help of a calculator are useful for investigating limits. Investigating a Limit Using a Table Investigate: ( ) → x lim 5 x 3 2 Solution EXAMPLE 1 Here ( ) = f x x5 2 and = c 3. Choose a value for x close to 3, such as 2.9.Then select additional numbers that are closer to 3 but remain less than 3. Next choose values of x greater than 3, such as 3.1, that get closer to 3. Table 1 shows the value of f for several choices, using a TI-84 Plus CE graphing utility. From Table 1, as x gets closer to 3, the value of ( ) = f x x5 2 appears to get closer to 45. This suggests that ( ) = → x lim 5 45 x 3 2 When choosing the values of x in a table, the number to start with and the subsequent entries are arbitrary. However, the entries should be chosen so that the table makes it clear what number the value of f is approaching. Now Work PROBLEM 7 Table 1
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