CWA CH 3 FINAL_97_12 - Chapter 3 THE DERIVATIVE 3.1 Limits...

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Chapter 3 THE DERIVATIVE 3.1 Limits 1. Since lim x ! 2 ¡ f ( x ) does not equal lim x ! 2 + f ( x ) , lim x ! 2 f ( x ) does not exist. The answer is c. 2. Since lim x ! 2 ¡ f ( x ) = lim x ! 2 + f ( x ) = ¡ 1 ; lim x ! 2 f ( x ) = ¡ 1 . The answer is a. 3. Since lim x ! 4 ¡ f ( x ) = lim x ! 4 + f ( x ) = 6 , lim x ! 4 f ( x ) = 6 . The answer is b. 4. Since lim x ! 1 ¡ f ( x ) = lim x ! 1 + f ( x ) = ¡1 ; lim x ! 1 f ( x ) = ¡1 . The answer is b. 5. (a) By reading the graph, as x gets closer to 3 from the left or right, f ( x ) gets closer to 3. lim x ! 3 f ( x ) = 3 (b) By reading the graph, as x gets closer to 0 from the left or right, f ( x ) gets closer to 1. lim x ! 0 f ( x ) = 1 6. (a) By reading the graph, as x gets closer to 2 from the left or right, F ( x ) gets closer to 4. lim x ! 2 F ( x ) = 4 (b) By reading the graph, as x gets closer to ¡ 1 from left or right, F ( x ) gets closer to 4. lim x 1 F ( x ) = 4 7. (a) By reading the graph, as x gets closer to 0 from the left or right, f ( x ) gets closer to 0. lim x ! 0 f ( x ) = 0 (b) By reading the graph, as x gets closer to 2 from the left, f ( x ) gets closer to ¡ 2 , but as x gets closer to 2 from the right, f ( x ) gets closer to 1. lim x ! 2 f ( x ) does not exist. 8. (a) By reading the graph, as x gets closer to 3 from the left or right, g ( x ) gets closer to 2. lim x ! 3 g ( x ) = 2 (b) By reading the graph, as x gets closer to 5 from the left, g ( x ) gets closer to ¡ 2 , but as x gets closer to 5 from the right, g ( x ) gets closer to 1. lim x ! 5 g ( x ) does not exist. 9. (a) (i) By reading the graph, as x gets closer to ¡ 2 from the left, f ( x ) gets closer to ¡ 1 : lim x 2 ¡ f ( x ) = ¡ 1 (ii) By reading the graph, as x gets closer to ¡ 2 from the right, f ( x ) gets closer to ¡ 1 2 : lim x 2 + f ( x ) = ¡ 1 2 (iii) Since lim x 2 ¡ f ( x ) = ¡ 1 and lim x 2 + f ( x ) = ¡ 1 2 ; lim x 2 f ( x ) does not exist. (iv) f ( ¡ 2) does not exist since there is no point on the graph with an x -coordinate of ¡ 2 : (b) (i) By reading the graph, as x gets closer to ¡ 1 from the left, f ( x ) gets closer to ¡ 1 2 : lim x 1 ¡ f ( x ) = ¡ 1 2 (ii) By reading the graph, as x gets closer to ¡ 1 from the right, f ( x ) gets closer to ¡ 1 2 : lim x 1 + f ( x ) = ¡ 1 2 (iii) Since lim x 1 ¡ f ( x ) = ¡ 1 2 and lim x 1 + f ( x ) = ¡ 1 2 ; lim x 1 f ( x ) = ¡ 1 2 : (iv) f ( ¡ 1) = ¡ 1 2 since ¡ ¡ 1 ; ¡ 1 2 ¢ is a point of the graph. 176
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Section 3.1 Limits 177 10. (a) (i) By reading the graph, as x gets closer to 1 from the left, f ( x ) gets closer to 1. lim x ! 1 ¡ f ( x ) = 1 (ii) By reading the graph, as x gets closer to 1 from the right, f ( x ) gets closer to 1. lim x ! 1 + f ( x ) = 1 (iii) Since lim x ! 1 ¡ f ( x ) = 1 and lim x ! 1 + f ( x ) = 1 ; lim x ! 1 f ( x ) = 1 : (iv) f (1) = 2 since (1 ; 2) is part of the graph. (b) (i) By reading the graph, as x gets closer to 2 from the left, f ( x ) gets closer to 0. lim x ! 2 ¡ f ( x ) = 0 (ii) By reading the graph, as x gets closer to 2 from the right, f ( x ) gets closer to 0. lim x ! 2 + f ( x ) = 0 (iii) Since lim x ! 2 ¡ f ( x ) = 0 and lim x ! 2 + f ( x ) = 0 ; lim x ! 2 f ( x ) = 0 : (iv) f (2) = 0 since (2 ; 0) is point of the graph. 11. By reading the graph, as x moves further to the right, f ( x ) gets closer to 3. Therefore, lim x !1 f ( x ) = 3 : 12. By reading the graph, as x moves further to the left, g ( x ) gets larger and larger. Therefore, lim x !¡1 g ( x ) = 1 : 13. lim x ! 2 F ( x ) in Exercise 6 exists because lim x ! 2 ¡ F ( x ) = 4 and lim x ! 2 + F ( x ) = 4 : lim x 2 f ( x )
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