ISM_T11_C02_C - Section 2.6 Continuity 3 Continuous on[c1 3 4 No discontinuous at x 1 1.5 lim k(x lim k(x 5(a Yes(c Yes 6(a Yes f(1 1(c No 7(a No

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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) œ œÁœ ! x x Ä" 5. (a) Yes (b) Yes, lim f(x) 0 x ı" œ (c) Yes (d) Yes 6. (a) Yes, f(1) 1 (b) Yes, 2 œœ x1 Ä (c) No (d) No 7. (a) No (b) No 8 . [ ) () ±"ß! ² !ß" ² "ß# ² #ß$ 9. f(2) 0, since 2(2) 4 0 ± ³ œ œ x x Ä# 10. f(1) should be changed to 2 œ Ä 11. Nonremovable discontinuity at x 1 because lim f(x) fails to exist ( lim f(x) 1 and 0). œ x x Ä Removable discontinuity at x 0 by assigning the number 0 to be the value of f(0) rather than x Ä! f(0) 1. œ 12. Nonremovable discontinuity at x 1 because 2 and 1). œ x x Ä Removable discontinuity at x 2 by assigning the number 1 to be the value of f(2) rather than x f(2) 2. œ 13. Discontinuous only when x 2 0 x 2 14. Discontinuous only when (x 2) 0 x 2 ±œÊœ ³ œÊœ ± # 15. Discontinuous only when x x (x 3)(x 1) 0 x 3 or x 1 # ±% ³$œ! Ê ± ± œ Ê œ œ 16. Discontinuous only when x 3x 10 0 (x 5)(x 2) 0 x 5 or x 2 # ±± œÊ ± ³œÊœ œ ± 17. Continuous everywhere. ( x 1 sin x defined for all x; limits exist and are equal to function values.) kk ±³ 18. Continuous everywhere. ( x 0 for all x; limits exist and are equal to function values.) ³"Á 19. Discontinuous only at x 0 œ 20. Discontinuous at odd integer multiples of , i.e., x = (2n ) , n an integer, but continuous at all other x. 11 ## ±" 21. Discontinuous when 2x is an integer multiple of , i.e., 2x n , n an integer x , n an integer, but œÊ œ n 1 # continuous at all other x. 22. Discontinuous when is an odd integer multiple of , i.e., (2n 1) , n an integer x 2n 1, n an 1 1 xx # # œ± Ê œ ± integer (i.e., x is an odd integer). Continuous everywhere else.
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100 Chapter 2 Limits and Continuity 23. Discontinuous at odd integer multiples of , i.e., x = (2n 1) , n an integer, but continuous at all other x. 11 ## ± 24. Continuous everywhere since x 1 1 and sin x 1 0 sin x 1 1 sin x 1; limits exist %# # ²  ± Ÿ Ê Ÿ Ÿ Ê ²   and are equal to the function values. 25. Discontinuous when 2x 3 0 or x continuous on the interval . ²³ ³± Ê ±ß_ 33 ± 26. Discontinuous when 3x 1 0 or x continuous on the interval . ±³ ³ Ê ß_ "" ± 27. Continuous everywhere: (2x 1) is defined for all x; limits exist and are equal to function values. ± "Î$ 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± œœ œ 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 # # 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 y1 ÄÄ Ä a b a b ab # # ±± œ ± œ ± œ " ± " sec 0 1, , and function continuous at y . œ " 32. lim tan cos sin x tan cos (sin(0)) tan cos (0) tan 1, and function continuous at x . x ±‘ ± ˆ ˆ 1 1 44 4 4 "Î$ œ œ œ !
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C02_C - Section 2.6 Continuity 3 Continuous on[c1 3 4 No discontinuous at x 1 1.5 lim k(x lim k(x 5(a Yes(c Yes 6(a Yes f(1 1(c No 7(a No

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