<strong>Sullivan</strong> AP˙<strong>Sullivan</strong>˙Chapter01 October 8, 2016 17:4 <strong>Sullivan</strong> 150 Chapter 1 • Limits and Continuity Figures 68 and 69 illustrate limits at infinity. y y 5 f (x) y y 5 f (x) L 1 L L 2 L 1 L L 2 M x . M x x , N N x lim f(x) 5 L x→∞ Figure 68 For any ε>0, there is a positive number M so that whenever x > M, then | f (x) − L| 0, there is a negative number N so that whenever x < N, then | f (x)−L| 0, suppose there is a δ>0 so that whenever 0 < |x − c| < δ, then | f (x) − L|
<strong>Sullivan</strong> AP˙<strong>Sullivan</strong>˙Chapter01 October 8, 2016 17:4 Section 1.6 • Assess Your Understanding 151 6. True or False A function f has a limit L at infinity, if for any given ε>0, there is a positive number M so that whenever x > M, then | f (x) − L| > ε. Skill Building In Problems 7–12, for each limit, find the largest δ that “works” for the given ε. 7. lim x→1 (2x) = 2, ε = 0.01 8. lim x→2 (−3x) =−6, ε = 0.01 9. lim(6x − 1) = 11 10. lim (2 − 3x) = 11 x→2 x→−3 ε = 1 ε = 1 2 3 ( 11. lim − 1 ) ( x→2 2 x + 5 = 4 12. lim 3x + 1 ) = 3 x→ 5 2 6 ε = 0.01 ε = 0.3 13. For the function f (x) = 4x − 1, we have lim x→3 f (x) = 11. For each ε>0, find a δ>0 so that whenever 0 < |x − 3| 0, find a δ>0 so that whenever 0 < |x + 2| 0, find a δ>0 so that ∣ whenever 0 < |x + 3| 0, find a δ>0 so that ∣ whenever 0 < |x − 2| 0, let δ ≤ . See TSM for (c) d ≤ 0.00025 complete proof. 2 ε (d) δ ≤ ε 4 20. Given any e > 0, let δ ≤ . See TSM for complete proof. 3 } . ε 21. Given any e > 0, let δ ≤ . See TSM for complete proof. 5 ε 22. Given any e > 0, let δ ≤ . See TSM for complete proof. 2 ⎧ ε ⎫ 23. Given any e > 0, let δ ≤ min⎨1, ⎬ . See TSM for complete proof. ⎩ 3 ⎭ ⎧ ε ⎫ 24. Given any e > 0, let δ ≤ min⎨1, ⎬ . See TSM for complete proof. ⎩ 4 ⎭ ⎧ ε ⎫ 25. Given any e > 0, let δ ≤ min⎨1, 2 ⎬ . See TSM for complete proof. ⎩ 7 ⎭ ⎧ ε ⎫ 26. Given any e > 0, let δ ≤ min⎨1, 15 ⎬ . See TSM for complete proof. ⎩ 4 ⎭ 27. Given any e > 0, let δ ≤ ε 3 . See TSM for complete proof. 28. Given any e > 0, let d ≤ e. See TSM for complete proof. ⎧ ε ⎫ 29. Given any e > 0, let δ ≤ min⎨1, ⎬ . See TSM for complete proof. ⎩ 3 ⎭ ⎧ ε ⎫ 30. Given any e > 0, let δ ≤ min⎨1, ⎬ . See TSM for complete proof. ⎩ 19 ⎭ 31. Given any e > 0, let d ≤ min{1, 6e}. See TSM for complete proof. ⎧ ε ⎫ 32. Given any e > 0, let δ ≤ min⎨1, 4 ⎬ . See TSM for complete proof. ⎩ 5 ⎭ 33. See TSM. 34. See TSM. 35. Given any e > 0, let ⎧ ε ⎫ δ ≤ min⎨1, 234 ⎬. ⎩ 7 ⎭ See TSM for complete proof. ⎧ ε ⎫ 36. Given any e > 0, let δ ≤ min⎨1, ⎬ . ⎩ 5 See TSM for complete proof. ⎭ 37. Given any e > 0, let d ≤ min{1, 26e}. See TSM for complete proof. 38. See TSM. ε 39. Given any e > 0, let δ ≤ . See TSM for complete proof. 1 + | m| 40. Given any e > 0, let ⎧ ε ⎫ δ ≤ min⎨ 1 ⎬ ⎩ 3 , 3 . See TSM. 13 ⎭ 41. x must be within 0.05 of 3. 42. x must be within approx. 0.087 of 3. 43. See TSM. 1 44. Given any e > 0, let N ≤− . See TSM for complete proof. ε 1 45. Given any e > 0, let M ≥ 2 ε . See TSM for complete proof. 46. N =− 10 47. See TSM. Section 1.6 • Assess Your Understanding 151 TE_<strong>Sullivan</strong>_Chapter01_PART II.indd 34 11/01/17 9:56 am
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