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ISM_T11_C02_C - Section 2.6 Continuity 3 Continuous on[c1 3...

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Section 2.6 Continuity 99 3. Continuous on [ 1 3] ß 4. No, discontinuous at x 1, 1.5 lim k(x) lim k(x) œ œ Á œ ! x x Ä " Ä " 5. (a) Yes (b) Yes, lim f(x) 0 x Ä " œ (c) Yes (d) Yes 6. (a) Yes, f(1) 1 (b) Yes, lim f(x) 2 œ œ x 1 Ä (c) No (d) No 7. (a) No (b) No 8. [ ) ( ) ( ) ( ) "ß ! !ß " "ß # #ß $ 9. f(2) 0, since lim f(x) 2(2) 4 0 lim f(x) œ œ œ œ x x Ä # Ä # 10. f(1) should be changed to 2 lim f(x) œ x 1 Ä 11. Nonremovable discontinuity at x 1 because lim f(x) fails to exist ( lim f(x) 1 and lim f(x) 0). œ œ œ x 1 x x Ä Ä " Ä " Removable discontinuity at x 0 by assigning the number lim f(x) 0 to be the value of f(0) rather than œ œ x Ä ! f(0) 1. œ 12. Nonremovable discontinuity at x 1 because lim f(x) fails to exist ( lim f(x) 2 and lim f(x) 1). œ œ œ x 1 x x Ä Ä " Ä " Removable discontinuity at x 2 by assigning the number lim f(x) 1 to be the value of f(2) rather than œ œ x Ä # f(2) 2. œ 13. Discontinuous only when x 2 0 x 2 14. Discontinuous only when (x 2) 0 x 2 œ Ê œ œ Ê œ # 15. Discontinuous only when x x (x 3)(x 1) 0 x 3 or x 1 # % $ œ ! Ê œ Ê œ œ 16. Discontinuous only when x 3x 10 0 (x 5)(x 2) 0 x 5 or x 2 # œ Ê œ Ê œ œ 17. Continuous everywhere. ( x 1 sin x defined for all x; limits exist and are equal to function values.) k k 18. Continuous everywhere. ( x 0 for all x; limits exist and are equal to function values.) k k " Á 19. Discontinuous only at x 0 œ 20. Discontinuous at odd integer multiples of , i.e., x = (2n ) , n an integer, but continuous at all other x. 1 1 # # " 21. Discontinuous when 2x is an integer multiple of , i.e., 2x n , n an integer x , n an integer, but 1 1 œ Ê œ n 1 # continuous at all other x. 22. Discontinuous when is an odd integer multiple of , i.e., (2n 1) , n an integer x 2n 1, n an 1 1 1 1 x x # # # # œ Ê œ integer (i.e., x is an odd integer). Continuous everywhere else.
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100 Chapter 2 Limits and Continuity 23. Discontinuous at odd integer multiples of , i.e., x = (2n 1) , n an integer, but continuous at all other x. 1 1 # # 24. Continuous everywhere since x 1 1 and sin x 1 0 sin x 1 1 sin x 1; limits exist % # #   " Ÿ Ÿ Ê Ÿ Ÿ Ê   and are equal to the function values. 25. Discontinuous when 2x 3 0 or x continuous on the interval . Ê ß _ 3 3 # # 26. Discontinuous when 3x 1 0 or x continuous on the interval . Ê ß _ " " 3 3 27. Continuous everywhere: (2x 1) is defined for all x; limits exist and are equal to function values. "Î$ 28. Continuous everywhere: (2 x) is defined for all x; limits exist and are equal to function values. "Î& 29. lim sin (x sin x) sin ( sin ) sin ( 0) sin 0, and function continuous at x . x Ä 1 œ œ œ œ œ 1 1 1 1 1 30. lim sin cos (tan t) sin cos (tan (0)) sin cos (0) sin 1, and function continuous at t . t Ä ! ˆ ˆ ˆ ˆ ‰ 1 1 1 1 # # # # œ œ œ œ œ ! 31. lim sec y sec y tan y 1 lim sec y sec y sec y lim sec (y 1) sec y sec ( ) sec 1 y 1 y 1 y 1 Ä Ä Ä a b a b a b a b # # # # # # œ œ œ " " sec 0 1, , and function continuous at y . œ œ œ " 32. lim tan cos sin x tan cos (sin(0)) tan cos (0) tan 1, and function continuous at x .
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