942 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function Investigating a Limit Using a Table Investigate: (a) − − → x x lim 4 2 x 2 2 (b) ( ) + → x lim 2 x 2 Solution EXAMPLE 2 (a) Here ( ) = − − f x x x 4 2 2 and = c 2. Notice that the domain of f is { } ≠ x x 2 , so f is not defined at 2. Choose values of x close to 2 on both sides and evaluate f at each choice, as shown in Table 2. The table suggests that as x gets closer to 2, the value of ( ) = − − f x x x 4 2 2 gets closer to 4. That is, − − = → x x lim 4 2 4 x 2 2 (b) Here ( ) = + g x x 2 and = c 2. The domain of g is all real numbers. See Table 3. The table suggests that as x gets closer to 2, the value of ( ) g x gets closer to 4. That is, ( ) + = → x lim 2 4 x 2 Table 3 Table 2 The conclusion that ( ) + = → x lim 2 4 x 2 could be obtained without Table 3; as x gets closer to 2, it follows that +x 2 gets closer to + = 2 2 4. Also, for part (a), you are correct if you make the observation that for ≠ x 2, we can divide out the factor ( ) −x 2 . ( ) ( )( ) = − − = − + − = + ≠ f x x x x x x x x 4 2 2 2 2 2 2 2 Therefore, ( ) − − = + = → → x x x lim 4 2 lim 2 4 x x 2 2 2 Let’s look at an example for which the factoring technique used above does not work. Investigating a Limit Using a Table Investigate: → x x lim sin x 0 EXAMPLE 3 Solution First, observe that the domain of the function ( ) = f x x x sin is { } ≠ x x 0 . Create Table 4, where x is measured in radians. Table 4 suggests that = → x x lim sin 1. x 0 Table 4
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