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Strategies to all odd-numbered textual content routines in Chapters P-10.
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Additional info for Calculus: Study And Solutions Guide, 8th Edition (3 volume set)
True; if f ͑x͒ ϭ g͑x͒, x c, then lim f ͑x͒ ϭ lim g͑x͒ and x→c x→c 103. False; f ͑1͒ is not defined and lim f ͑x͒ does not exist. x→1 at least one of these limits (if they exist) does not equal the corresponding function at x ϭ c. 105. (a) f ͑x͒ ϭ Άb 0 ≤ x < b b < x ≤ 2b 0 (b) g͑x͒ ϭ y 2b Ά x 2 0 ≤ x ≤ b bϪ x 2 b < x ≤ 2b y 2b b x b b 2b NOT continuous at x ϭ b. x b 2b Continuous on ͓0, 2b͔. 41 42 Chapter 1 107. f ͑x͒ ϭ Limits and Their Properties Ίx ϩ c2 Ϫ c x , c > 0 Domain: x ϩ c2 ≥ 0 ⇒ x ≥ Ϫc2 and x Ίx ϩ c2 Ϫ c lim x x→0 ϭ lim Ίx ϩ c2 Ϫ c x→0 x 0, ͓Ϫc2, 0͒ ʜ ͑0, ϱ͒ и Ίx ϩ c2 ϩ c Ίx ϩ c2 ϩ c ͑x ϩ c2͒ Ϫ c2 1 1 ϭ lim ϭ x→0 x͓Ίx ϩ c2 ϩ c͔ x→0 Ίx ϩ c2 ϩ c 2c ϭ lim Define f ͑0͒ ϭ 1͑͞2c͒ to make f continuous at x ϭ 0.
X2 ϩ 1 41. f ͑x͒ ϭ xϩ2 ͑x ϩ 2͒͑x Ϫ 5͒ has a nonremovable discontinuity at x ϭ 5 since lim f ͑x͒ x→5 does not exist, and has a removable discontinuity at x ϭ Ϫ2 since lim f ͑x͒ ϭ lim x→Ϫ2 43. f ͑x͒ ϭ Խx ϩ 2Խ has a nonremovable discontinuity at x ϭ Ϫ2 since 45. f ͑x͒ ϭ Άx,x , xϩ2 x→Ϫ2 1 1 ϭϪ . xϪ5 7 lim f ͑x͒ does not exist. x→Ϫ2 x ≤ 1 x > 1 2 has a possible discontinuity at x ϭ 1. 1. f ͑1͒ ϭ 1 2. lim f ͑x͒ ϭ limϪ x ϭ 1 x→1Ϫ x→1 x→1ϩ x→1 · lim f ͑x͒ ϭ 1 lim f ͑x͒ ϭ limϩ x2 ϭ 1 x→1 3. f ͑1͒ ϭ lim f ͑x͒ x→1 f is continuous at x ϭ 1, therefore, f is continuous for all real x.
X ϭ lim Ϫ x→3 lim x→Ϫ3 Ϫ without bound as x → Ϫ3 Ϫ . 1 1 Ϫ x Ϫ ͑x ϩ ⌬x͒ x ϩ ⌬x x ϭ lim Ϫ 13. lim Ϫ ⌬x→0 ⌬x→0 ⌬x x͑x ϩ ⌬x͒ 15. limϪ f ͑x͒ ϭ limϪ The function is NOT continuous at x ϭ 4. The function is NOT continuous at x ϭ 3. xϪ5 1 1 ϭ lim ϭ x2 Ϫ 25 x→5ϩ x ϩ 5 10 ԽxԽ ϭ lim f ͑x͒ ϭ 2 x→4 ϩ (b) limϪ f ͑x͒ ϭ Ϫ2 x→3 (c) lim f ͑x͒ ϭ 1 11. limϪ 5. (a) x→3 x→3 x→5 37 Continuity and One-Sided Limits 1. (a) limϩ f ͑x͒ ϭ 1 7. limϩ Continuity and One-Sided Limits Ϫ1 x͑x ϩ ⌬x͒ Ϫ1 1 ϭϪ 2 x͑x ϩ 0͒ x xϩ2 5 ϭ 2 2 17.