1. 1.5 and 1.6 – Limits and Continuity
Math 140
1.5 and 1.6 – Limits and Continuity

2. Let’s warm up by visually plugging inputs into a function and reading off the outputs.

3. Ex 1. Find the following, given the graph of the crazy piecewise-defined function, 𝑓(𝑥), above.

4. The limit of a function is a core concept in calculus
The limit of a function is a core concept in calculus. Here, you have to imagine an animation of the inputs and outputs of a function. Ask: what’s happening to the outputs as the inputs are changing?

5. Note: 𝒙→ 𝟏 − means “𝒙 approaches 1 from the left” 𝒙→ 𝟏 + means “𝒙 approaches 1 from the right”

6. Ex 2. Find the following limits, given the graph of 𝑓(𝑥) above.

7. We can use +∞ and −∞ to describe the behavior of the following function.

8. Ex 3. Find the following limits, given the graph of 𝑓(𝑥) above.

9. Limits with 𝑥→ 𝑐 − or 𝑥→ 𝑐 + are called “one-sided” limits, because the inputs (𝑥-values) are approaching the number 𝑐 from one side (either the left or right side).

10. In the future, we’ll mostly use the “regular” limit, which requires the left- and right-hand limits to exist and be equal. The “regular” limit does not have a “-” or “+”, and is just written using 𝑥→𝑐.

11. In the future, we’ll mostly use the “regular” limit, which requires the left- and right-hand limits to exist and be equal. The “regular” limit does not have a “-” or “+”, and is just written using 𝑥→𝑐.

12. Note: lim 𝑥→𝑐 𝑓(𝑥) =𝐿 if an only if lim 𝑥→ 𝑐 − 𝑓(𝑥) = lim 𝑥→ 𝑐 + 𝑓(𝑥) =𝐿

13. As with one-sided limits, regular limits only care about the behavior of the function near 𝑥=𝑐, not at 𝑥=𝑐.

14. Ex 4. Find the following limits, given the graph of 𝑓(𝑥) above.

15. Algebraic Limits
Let’s transition from graphs and visualizations to determining limits algebraically.

16. Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________.

17. 𝒌
Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________.

18. 𝒌 𝒄
Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________.

19. 𝒌 𝒄 Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________.
ex: lim 𝑥→71 3 = ______ lim 𝑥→42 𝑥 = ______

20. 𝒌 𝒄 𝟑 Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________.
ex: lim 𝑥→71 3 = ______ lim 𝑥→42 𝑥 = ______

21. 𝒌 𝒄
Note: In general, lim 𝑥→𝑐 𝑘 = ______ and lim 𝑥→𝑐 𝑥 = ________. 𝟑 𝟒𝟐
ex: lim 𝑥→71 3 = ______ lim 𝑥→42 𝑥 = ______

22. To handle limits algebraically, it will also help to have the following properties.

23. Properties of Limits
If lim 𝑥→𝑐 𝑓(𝑥) and lim 𝑥→𝑐 𝑔(𝑥) both exist, then the following properties are true: 𝟏. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 𝟐. 𝐥𝐢𝐦 𝒙→𝒄 𝒌𝒇(𝒙) =𝒌 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) (ℎ𝑒𝑟𝑒 𝑘 is a constant) 𝟑. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 𝟒. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 (if lim 𝑥→𝑐 𝑔(𝑥) ≠0) 𝟓. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 (if lim 𝑥→𝑐 𝑓 𝑥 𝑝 exists)

24. Properties of Limits
If lim 𝑥→𝑐 𝑓(𝑥) and lim 𝑥→𝑐 𝑔(𝑥) both exist, then the following properties are true: 𝟏. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 𝟐. 𝐥𝐢𝐦 𝒙→𝒄 𝒌𝒇(𝒙) =𝒌 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) (ℎ𝑒𝑟𝑒 𝑘 is a constant) 𝟑. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 𝟒. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 (if lim 𝑥→𝑐 𝑔(𝑥) ≠0) 𝟓. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 (if lim 𝑥→𝑐 𝑓 𝑥 𝑝 exists)

25. Properties of Limits
If lim 𝑥→𝑐 𝑓(𝑥) and lim 𝑥→𝑐 𝑔(𝑥) both exist, then the following properties are true: 𝟏. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 𝟐. 𝐥𝐢𝐦 𝒙→𝒄 𝒌𝒇(𝒙) =𝒌 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) (ℎ𝑒𝑟𝑒 𝑘 is a constant) 𝟑. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 𝟒. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 (if lim 𝑥→𝑐 𝑔(𝑥) ≠0) 𝟓. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 (if lim 𝑥→𝑐 𝑓 𝑥 𝑝 exists)

26. Properties of Limits
If lim 𝑥→𝑐 𝑓(𝑥) and lim 𝑥→𝑐 𝑔(𝑥) both exist, then the following properties are true: 𝟏. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 𝟐. 𝐥𝐢𝐦 𝒙→𝒄 𝒌𝒇(𝒙) =𝒌 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) (ℎ𝑒𝑟𝑒 𝑘 is a constant) 𝟑. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 𝟒. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 (if lim 𝑥→𝑐 𝑔(𝑥) ≠0) 𝟓. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 (if lim 𝑥→𝑐 𝑓 𝑥 𝑝 exists)

27. Properties of Limits
If lim 𝑥→𝑐 𝑓(𝑥) and lim 𝑥→𝑐 𝑔(𝑥) both exist, then the following properties are true: 𝟏. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 𝟐. 𝐥𝐢𝐦 𝒙→𝒄 𝒌𝒇(𝒙) =𝒌 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) (ℎ𝑒𝑟𝑒 𝑘 is a constant) 𝟑. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 𝟒. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 (if lim 𝑥→𝑐 𝑔(𝑥) ≠0) 𝟓. 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒑 (if lim 𝑥→𝑐 𝑓 𝑥 𝑝 exists)

28. Note: All of the above properties work with 𝑥→ 𝑐 − and 𝑥→ 𝑐 + .

29. Ex 5. Use the properties of limits to find the following
Ex 5. Use the properties of limits to find the following. lim 𝑥→−2 (2 𝑥 3 − 𝑥 2 +3)

30. Note: If 𝑝(𝑥) and 𝑞(𝑥) are polynomials, then…  lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐)  lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 (if 𝑞 𝑐 ≠0) Ex 6. Find lim 𝑥→−1 2 𝑥 4 −𝑥+1 𝑥−1

31. Note: If 𝑝(𝑥) and 𝑞(𝑥) are polynomials, then…  lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐)  lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 (if 𝑞 𝑐 ≠0) Ex 6. Find lim 𝑥→−1 2 𝑥 4 −𝑥+1 𝑥−1

32. Note: If 𝑝(𝑥) and 𝑞(𝑥) are polynomials, then…  lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐)  lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 (if 𝑞 𝑐 ≠0) Ex 6. Find lim 𝑥→−1 2 𝑥 4 −𝑥+1 𝑥−1

33. What happens if the bottom polynomial is 0 at 𝑥=𝑐
What happens if the bottom polynomial is 0 at 𝑥=𝑐? If the top is also 0, look for a factor to cancel. Ex 7. Find lim 𝑥→3 𝑥 2 −4𝑥+3 𝑥 2 −5𝑥+6

34. What happens if the bottom polynomial is 0 at 𝑥=𝑐
What happens if the bottom polynomial is 0 at 𝑥=𝑐? If the top is also 0, look for a factor to cancel. Ex 7. Find lim 𝑥→3 𝑥 2 −4𝑥+3 𝑥 2 −5𝑥+6

35. What happens if the bottom polynomial is 0 at 𝑥=𝑐
What happens if the bottom polynomial is 0 at 𝑥=𝑐? If the top is also 0, look for a factor to cancel. Ex 7. Find lim 𝑥→3 𝑥 2 −4𝑥+3 𝑥 2 −5𝑥+6

36. What happens if the bottom polynomial is 0 at 𝑥=𝑐
What happens if the bottom polynomial is 0 at 𝑥=𝑐? If the top is also 0, look for a factor to cancel. Ex 7. Find lim 𝑥→3 𝑥 2 −4𝑥+3 𝑥 2 −5𝑥+6

37. Ex 8. Find lim 𝑥→4 𝑥 −2 𝑥−4

38. Ex 8. Find lim 𝑥→4 𝑥 −2 𝑥−4

39. What if you can’t cancel a factor and the bottom polynomial is still 0 at 𝑥=𝑐? Check if the left- and right-side limits are the same. If they’re both, for example, +∞, then the regular limit is +∞. If they’re different, then the regular limit DNE.

40. What if you can’t cancel a factor and the bottom polynomial is still 0 at 𝑥=𝑐? Check if the left- and right-side limits are the same. If they’re both, for example, +∞, then the regular limit is +∞. If they’re different, then the regular limit DNE.

41. Ex 9. lim 𝑥→0 1 𝑥 2 Ex 10. lim 𝑥→0 1 𝑥

42. Ex 9. lim 𝑥→0 1 𝑥 2 Ex 10. lim 𝑥→0 1 𝑥

43. Ex 11. Find lim 𝑥→2 𝑥−1 𝑥−2

44. Ex 11. Find lim 𝑥→2 𝑥−1 𝑥−2

45. Here are some limits with a piecewise-defined function. Ex 12
Here are some limits with a piecewise-defined function. Ex 12. Suppose 𝑓 𝑥 = 2− 𝑥 2 3𝑥−3 if−2≤𝑥<1 if 𝑥≥1 lim 𝑥→ 1 − 𝑓(𝑥) = lim 𝑥→ 1 + 𝑓(𝑥) = lim 𝑥→1 𝑓(𝑥)

46. Here are some limits with a piecewise-defined function. Ex 12
Here are some limits with a piecewise-defined function. Ex 12. Suppose 𝑓 𝑥 = 2− 𝑥 2 3𝑥−3 if−2≤𝑥<1 if 𝑥≥1 lim 𝑥→ 1 − 𝑓(𝑥) = lim 𝑥→ 1 + 𝑓(𝑥) = lim 𝑥→1 𝑓(𝑥)

47. Limits at Infinity
Above, the two horizontal asymptotes are _________ and ________.

48. Limits at Infinity
Above, the two horizontal asymptotes are _________ and ________.

49. Limits at Infinity
Above, the two horizontal asymptotes are _________ and ________.

50. Limits at Infinity
Above, the two horizontal asymptotes are _________ and ________.

51. Ex 13. lim 𝑥→+∞ 1 𝑥 In general, for 𝑘>0, 𝐥𝐢𝐦 𝒙→∞ 𝟏 𝒙 𝒌 =𝟎 𝐥𝐢𝐦 𝒙→−∞ 𝟏 𝒙 𝒌 =𝟎

52. Ex 13. lim 𝑥→+∞ 1 𝑥 In general, for 𝑘>0, 𝐥𝐢𝐦 𝒙→∞ 𝟏 𝒙 𝒌 =𝟎 𝐥𝐢𝐦 𝒙→−∞ 𝟏 𝒙 𝒌 =𝟎

53. Ex lim 𝑥→∞ 𝑥 2 𝑥 2 +𝑥+1

54. Ex 14. lim 𝑥→∞ 3+2 𝑥 2 𝑥 2 +𝑥+1 Note: Same trick works when 𝑥→−∞.

55. Ex 14. lim 𝑥→∞ 3+2 𝑥 2 𝑥 2 +𝑥+1 Note: Same trick works when 𝑥→−∞.

56. Ex lim 𝑥→−∞ −2 𝑥 2 +4𝑥+3 𝑥−2

57. Limits Summary When taking lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 … 1
Limits Summary When taking lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 … 1. …if top→0 bot→0 then try to cancel factor. 2. …if top→nonzero bot→0 then you’ll have either +∞, −∞, or DNE. When taking lim 𝑥→±∞ 𝑝 𝑥 𝑞 𝑥 if top→±∞ bot→±∞ then divide top/bot by highest degree term of bottom.

58. A ______________________ function is one with no “holes” or “gaps”.

59. A ______________________ function is one with no “holes” or “gaps”.
continuous

60. These functions are not continuous at 𝑥=𝑐 because they have “holes”:

61. These functions are not continuous at 𝑥=𝑐 because they have “gaps”:

62. Here’s the formal definition:
𝑓 𝑥 is continuous at 𝒄 if lim 𝑥→𝑐 𝑓(𝑥) =𝑓(𝑐)
(Note that 𝑓(𝑐) must be defined, and lim 𝑥→𝑐 𝑓(𝑥) must exist.)

63. Ex 16. Is 𝑓 𝑥 = 𝑥 2 +3𝑥−7 continuous at 𝑥=2. Ex 17
Ex 16. Is 𝑓 𝑥 = 𝑥 2 +3𝑥−7 continuous at 𝑥=2? Ex 17. Is 𝑓 𝑥 = 𝑥−1 𝑥−2 continuous at 𝑥=2?

64. Ex 16. Is 𝑓 𝑥 = 𝑥 2 +3𝑥−7 continuous at 𝑥=2. Ex 17
Ex 16. Is 𝑓 𝑥 = 𝑥 2 +3𝑥−7 continuous at 𝑥=2? Ex 17. Is 𝑓 𝑥 = 𝑥−1 𝑥−2 continuous at 𝑥=2?

65. Notes: Polynomials and rational functions are continuous for all 𝑥 where the functions are defined. ex: 𝑓 𝑥 =5 𝑥 4 +6 𝑥 2 −𝑥+2 is continuous for all 𝑥 ex: 𝑓 𝑥 = 𝑥 2 −1 𝑥+1 is continuous for all 𝑥≠−1 Why? Since lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐), and lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 if 𝑞 𝑐 ≠0 if 𝑞 𝑐 ≠0

66. Notes: Polynomials and rational functions are continuous for all 𝑥 where the functions are defined. ex: 𝑓 𝑥 =5 𝑥 4 +6 𝑥 2 −𝑥+2 is continuous for all 𝑥 ex: 𝑓 𝑥 = 𝑥 2 −1 𝑥+1 is continuous for all 𝑥≠−1 Why? Since lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐), and lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 if 𝑞 𝑐 ≠0 if 𝑞 𝑐 ≠0

67. Notes: Polynomials and rational functions are continuous for all 𝑥 where the functions are defined. ex: 𝑓 𝑥 =5 𝑥 4 +6 𝑥 2 −𝑥+2 is continuous for all 𝑥 ex: 𝑓 𝑥 = 𝑥 2 −1 𝑥+1 is continuous for all 𝑥≠−1 Why? Since lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐), and lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 if 𝑞 𝑐 ≠0 if 𝑞 𝑐 ≠0

68. Notes: Polynomials and rational functions are continuous for all 𝑥 where the functions are defined. ex: 𝑓 𝑥 =5 𝑥 4 +6 𝑥 2 −𝑥+2 is continuous for all 𝑥 ex: 𝑓 𝑥 = 𝑥 2 −1 𝑥+1 is continuous for all 𝑥≠−1 Why? Since lim 𝑥→𝑐 𝑝(𝑥) =𝑝(𝑐), and lim 𝑥→𝑐 𝑝 𝑥 𝑞 𝑥 = 𝑝 𝑐 𝑞 𝑐 if 𝑞 𝑐 ≠0 if 𝑞 𝑐 ≠0

69. Ex 18. List all values of 𝑥 for which 𝑔 𝑥 = 1−𝑥 𝑥+2 if 𝑥<1 if 𝑥≥1 is not continuous.