University:
Bradley University
Course:
MTH 115 | Brief Calculus With Applications I
Academic year:

2020
Views:

315

Pages:

15
Author:

Gudrunecg9m

4/7/2020 1.4 Continuity & One-Sided Limits O The existence of a limit: Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim 𝑓(𝑥)=L and lim 𝑓(𝑥)=L 𝑥→𝑐− 𝑥→𝑐+ O lim 𝑓(𝑥)=L means approaches from 𝑥→𝑐+ the right O lim 𝑓(𝑥)=L means approaches from 𝑥→𝑐− the left 1 4/7/2020 O Definition of continuity on a closed Interval A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and lim 𝑓(𝑥)=f(a) and 𝑥→𝑎+ lim 𝑓(𝑥)=f(b) 𝑥→𝑏− The function f is continuous from the right at a and continuous from the left at b O Continuity at a point A function f is continuous at c if the following three conditions are met 1. f(c) is defined 2. lim 𝑓(𝑥) exists 𝑥→𝑐 3. lim 𝑓(𝑥) = f(c) 𝑥→𝑐 2 4/7/2020 O One Sided Limits lim 𝑓(𝑥)= L Limit approaches from the right lim 𝑓(𝑥)= L Limit approaches from the left lim 𝑓(𝑥)= L Limit approaches from the 𝑥→𝑐+ 𝑥→𝑐− 𝑥→𝑐 right and the left and must be equal to exist 3 4/7/2020 1. lim 𝑓(𝑥) 𝑥→2+ 2. lim 𝑓(𝑥) 𝑥→2− 3. lim 𝑓(𝑥) 𝑥→2 6. 7. 8. 4. f(2) 9. 5. 10. Continuous At 2? 1. lim 𝑓(𝑥) 𝑥→2+ 2. lim 𝑓(𝑥) 𝑥→2− 3. lim 𝑓(𝑥) 𝑥→2 4. f(2) 5. Continuous At 2? lim 𝑓(𝑥) 11. 12. lim 𝑓(𝑥) 13. lim 𝑓(𝑥) 𝑥→1+ 𝑥→1− 𝑥→1 f(1) 14. 15. Continuous at 1? lim 𝑓(𝑥) 𝑥→3+ lim 𝑓(𝑥) 𝑥→3− lim 𝑓(𝑥) 𝑥→3 f(3) Continuous at 3? 1. 3 2. 2 3. Does not exist 4. 2 5. No because the limit does not equal the function value 4 4/7/2020 6. 7. 8. 9. 10. lim 𝑓(𝑥) 𝑥→1+ lim 𝑓(𝑥) 𝑥→1− lim 𝑓(𝑥) 𝑥→1 f(1) Continuous at 1? 11. 12. 𝑥→3+ 13. lim 𝑓(𝑥) 14. 15. lim 𝑓(𝑥) lim 𝑓(𝑥) 𝑥→3− 𝑥→3 f(3) Continuous at 3? 1. 1 2. 1 3. 1 4. 2 5. No because the limit does not equal the function value 1. 1 2. 1 3. 1 4. 1 5. Yes because the limit equals the function value 5 4/7/2020 1. lim 𝑓(𝑥) 𝑥→−3+ 2. lim 𝑓(𝑥) 𝑥→−3− 3. lim 𝑓(𝑥) 𝑥→3 6. 7. 𝑥→0+ lim 𝑓(𝑥) 11. 12. lim 𝑓(𝑥) 13. lim 𝑓(𝑥) lim 𝑓(𝑥) 𝑥→−1+ 𝑥→−1− 8. 𝑥→−1 4. f(3) 9. f(-1) 5. 10. Continuous At 3? Continuous at -1? lim 𝑓(𝑥) lim 𝑓(𝑥) 𝑥→0− 𝑥→0 14. f(0) 15. Continuous at 0? O Tell if function is continuous at x = 4 O𝑓 𝑥 = 2𝑥 + 3, 𝑥 ≤ 4 7+ 16 ,𝑥 𝑥 >4 O f(4) = 2(4)+3=11 O lim 2𝑥 + 3 = 11 𝑥→4− O lim 7 + 𝑥→4+ 16 = 𝑥 11 O lim 𝑓(𝑥) = 11 therefore continuous at x 𝑥→4 =4 6 4/7/2020 O𝑓 𝑥 = 𝑥 + 1, 𝑥 ≤ 4 ` 𝑥 2 + 1, 𝑥 > 4 1. lim 𝑓(𝑥) 𝑥→4+ 2. lim 𝑓(𝑥) 𝑥→4− 6. 7. lim 𝑓(𝑥) 𝑥→0+ lim 𝑓(𝑥) 𝑥→0− 3. lim 𝑓(𝑥) 8. 4. f(4) 9. f(0) 10. Continuous at 0? 𝑥→4 5. Continuous at 4? lim 𝑓(𝑥) 𝑥→0 O Discuss the continuity: O 1. f(x) = 1 𝑥 Graph is undefined at x = 0; therefore graph is continuous everywhere except x = 0; nonremovable discontinuity 7 4/7/2020 Discontinuity O Removable Discontinuity O A discontinuity at c is removable if f can be made continuous by appropriately defining or redefining f(c). O Nonremovable Discontinuity O If the limit as x approaches f(x) does not exist, there is no way to fulfill the continuity test and discontinuity is nonremovable. 2. f(x) = 𝑥 2 −1 𝑥−1 O Use graphing calculator: O Change window to -5, 5, 1, -5, 5, 1 Discontinuous at x = 1 because denominator cannot equal 0; however, it is a removable discontinuity at x = 1 𝑥 2 −1 𝑥−1 = (𝑥−1)(𝑥+1) = 𝑥−1 x+1 8 4/7/2020 f(x) = sin x O Continuous function 3. 4. 𝑓 𝑥 = 1 − 𝑥 2 [– 1, 1] O Continuous from [– 1, 1] Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? 5. 𝑓 𝑥 = 𝑥 2 + 5𝑥 − 6 Continuous 6. 𝑥 f(x) = 𝑥2 −4 𝑥 (𝑥 − 2)(𝑥 + 2) Discontinuous at x = ±2; Nonremovable 7. 𝑥−4 f(x) = 𝑥2 −16 𝑥−4 (𝑥 − 4)(𝑥 + 4) Discontinuous at x = ±4; Nonremovable at x=– 4; Removable at x = 4 9 4/7/2020 O Find the constant of a such that the function is continuous on the entire real line. O𝑓 𝑥 = 3𝑥 + 𝑎 𝑥 > 4 𝑥 2, 𝑥 ≤ 4 O f(4) O = 𝑥 2 = 42 = 16 O lim 𝑓(𝑥) 𝑥→4− 2 O = 4 = 16 O lim 3𝑥 + 𝑎= 16 𝑥→4+ O 3x + a = 3(4) + a = 16; therefore a = 4 O Intermediate value theorem If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b) then there is at least on number c in [a, b] such that f(c) =k Example: f(x) = x3 – 1 [– 1, 2] 10 4/7/2020 Homework Answers 1. 2. 3. 4. 5. 6. 1,1,1 Continuous at 3 – 2, – 2, – 2, Continuous at – 2 0,0,0 NOT continuous at 3 2, 2, 2, NOT continuous at – 2 2, – 2, DNE, NOT continuous at 4 0, 2, DNE, NOT continuous at – 1 Homework Answers 7. 8. lim lim x 5 x 5 1 1 1     x 2  25 ( x  5)( x  5) x  5 5  5 10 2 x  ( x  2) 1 1 1     2 x  4 ( x  2)( x  2) x  2 2  2 4 11 4/7/2020 Homework Answers 9. DNE 10.¼ 11.– 1 12.1 13.– 1/x2 Homework Answers 15. 16. 17. 18. 19. 20. 25. 26. 5/2 27. Discontinuous 2,2,2 at each integer 2,2,2 28. Discontinuous 0 function value does DNE not equal limit value DNE at x = 1 The function is discontinuous at x = 2 and x = – 2 because there is no function value The function is discontinuous at x = – 1 because there is no function value 12 4/7/2020 Homework Page 79-80 33. Continuous 34. Continuous 39. 40. 35. Continuous 36. Continuous Continuous Removable at x=3 Nonremovable at x = –3 37. Removable at x = 0 Nonremovable at x = 1 38. Nonremovable at x = 1, x = – 1 Homework Page 79-80 41. Nonremovable at x = 5 Removable at x = –2 42. Nonremovable at x = –2 Removable at x = 1 43. Nonremovable at x = –2 44. Nonremovable at x = 3 45. Continuous 46. Continuous 47. Nonremovable discontinuity at x = 2 because the limit at x = 2 does not exist 13 4/7/2020 Homework Page 79-80 48. Nonremovable discontinuity at x = 2 because limit at x = 2 does not exist 49. Continuous 50. Continuous 51. Nonremovable at n Π 2 52.Nonremovable at 2n+1 Because asymptotes occur at Π+nΠ= Πx 2 2 Homework Page 79-80 57. 58. f(2) = 8 lim ax2 = 8 a(2) 2 = 8 a=2 f(0) = a lim f(x) = 4 a=4 14 4/7/2020 Bonus 60. ( x  a)( x  a)  xa xa lim (x + a) = 2a 2a = 8 a=4 15

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