Download Calculus: Study And Solutions Guide, 8th Edition (3 volume by Ron Larson, Robert P. Hostetler, Bruce H. Edwards PDF
By Ron Larson, Robert P. Hostetler, Bruce H. Edwards
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)
Example text
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.