SECTION 14.1 Investigating Limits Using Tables and Graphs 943 Using a Graphing Utility to Investigate a Limit Investigate: − + − − + − → x x x x x x lim 2 4 8 2 2 x 2 3 2 4 3 EXAMPLE 4 Solution Table 5 shows values of the rational expression for x close to 2.The table suggests that − + − − + − = → x x x x x x lim 2 4 8 2 2 0.889 x 2 3 2 4 3 rounded to three decimal places. Figure 1 y 5 f(x) f(c) not defined; lim f(x) 5 N (c) N x y x c y 5 f(x) f(c) ? N; lim f(x) 5 N (b) f(c) N x y x c y 5 f(x) N x c x x y x c x x c x f(c) 5 N; lim f(x) 5 N (a) x c Investigate a Limit by Graphing Investigate: ( ) → f x lim x 2 if ( ) = − ≠ = ⎧ ⎨ ⎪ ⎩⎪⎪ f x x x x 3 2 if 2 3 if 2 Solution EXAMPLE 5 The function f is a piecewise-defined function. Its graph is shown in Figure 2(a) drawn by hand. Figure 2(b) shows the graph using a TI-84 Plus CE graphing utility. Table 5 Now Work PROBLEM 43 2 Investigate a Limit Using a Graph The graph of a function f can also be of help in investigating limits. See Figure 1. In each graph, notice that as x gets closer to c, the value of f gets closer to the number N. This suggests that ( ) = → f x N lim x c This is the conclusion regardless of the value of f at c. In Figure 1(a), ( ) = f c N, and in Figure 1(b), ( ) ≠ f c N. Figure 1(c) suggests that ( ) = → f x N lim , x c even if f is not defined at c. Figure 2 2 4 6 –2 2 (a) 4 x y (b) Y1 = (3x 2 2)(x = 2) 1 3(x = 2) (2, 3) 8 0 0 4 (continued)
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