944 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function See Figure 3. By ZOOMing in around = x 2 and TRACEing, we conclude from the graph that ( ) = → f x lim 4. x 2 Figure 3 Notice in Example 4 that the value of f at 2, that is, ( ) = f 2 3, plays no role in the conclusion that ( ) = → f x lim 4. x 2 In fact, even if f were undefined at 2, it would still be the case that ( ) = → f x lim 4. x 2 Now Work PROBLEMS 17 AND 23 Sometimes there is no single number that the values of f approach as x gets closer to c . In this case, we say that f has no limit as x approaches c or that ( ) → f x lim x c does not exist . A Function That Has No Limit at 0 Investigate: ( ) → f x lim x 0 if ( ) = ≤ > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x if 0 1 if 0 Solution EXAMPLE 6 See Figure 4.As x gets closer to 0 but remains negative, the value of f also gets closer to 0. As x gets closer to 0 but remains positive, the value of f always equals 1. Since there is no single number that the values of f are close to when x is close to 0, we conclude that ( ) → f x lim x 0 does not exist. Now Work PROBLEM 37 In the next section, we will see how algebra can be used to obtain exact limits of functions. Figure 4 ( ) = ≤ > ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ f x x x x if 0 1 if 0 22 24 2 22 2 4 x y ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 14.1 Assess Your Understanding 1. Graph ( ) = − ≠ = ⎧ ⎨ ⎪ ⎩⎪⎪ f x x x x 3 2 if 2 3 if 2 (pp. 105–107) 2. If ( ) = ≤ > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x if 0 1 if 0 what is ( ) f 0 ? (pp. 105–107) 3. Multiple Choice The limit of a function ( ) = y f x as x approaches c is denoted by the symbol . (a) ( ) → f x lim x c (b) ( ) → f x lim f c (c) ( ) ← f x lim x c (d) ( ) → f x lim c f 4. If a function f does not approach a single number as x approaches c , then we say that ( ) → f x lim x c . Concepts and Vocabulary 5. True or False ( ) = → f x N lim x c may be described by saying that the value of ( ) f x gets closer to N as x gets closer to c but remains unequal to c . 6. True or False ( ) → f x lim x c exists and equals some number for any function f as long as c is in the domain of f. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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