1. §1.3: Properties of Limits with Trigonometry
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§1.3: Properties of Limits with Trigonometry

2. Evaluating Limits Analytically
Section 1.3
Calculus BC AP/Dual, Revised ©2017
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§1.3: Properties of Limits with Trigonometry

3. Easy Limits Start with direct substitution Simplify
𝐥𝐢𝐦 𝒙→𝒄 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕=𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝐥𝐢𝐦 𝒙→𝒄 𝒙=𝒄
𝐥𝐢𝐦 𝒙→𝒄 𝒙 𝒏 = 𝒄 𝒏
REMEMBER: IT IS WHAT 𝒙 APPROACHES NOT WHAT 𝒙 IS
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§1.3: Properties of Limits with Trigonometry

4. §1.3: Properties of Limits with Trigonometry
Example 1
Evaluate 𝐥𝐢𝐦 𝒙→𝟑 𝟐𝒙+𝟓
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§1.3: Properties of Limits with Trigonometry

5. §1.3: Properties of Limits with Trigonometry
Example 2
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙
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§1.3: Properties of Limits with Trigonometry

6. §1.3: Properties of Limits with Trigonometry
Example 3
Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙−𝟐
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§1.3: Properties of Limits with Trigonometry

7. Review of Compositions
Determine what is substituted
Take the INSIDE function and replace it
Take the outside function and bring it down
Replace the variable with the leftover variable
Simplify the expression
Notation: They may write it as 𝒇 𝒈 𝒙 or(𝒇 ○ 𝒈) 𝒙 . The meaning is the same.
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§1.3: Properties of Limits with Trigonometry

8. §1.3: Properties of Limits with Trigonometry
Review Example A
If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒙
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§1.3: Properties of Limits with Trigonometry

9. §1.3: Properties of Limits with Trigonometry
Review Example B
If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒈 𝒇 𝒙
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§1.3: Properties of Limits with Trigonometry

10. §1.3: Properties of Limits with Trigonometry
Example 4
Find the following limit given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 , determine 𝐥𝐢𝐦 𝒙→−𝟓 𝒈 𝒇 𝒙
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§1.3: Properties of Limits with Trigonometry

11. §1.3: Properties of Limits with Trigonometry
Your Turn
Find the following limit given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 , determine 𝐥𝐢𝐦 𝒙→−𝟓 𝒇 𝒈 𝒙
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§1.3: Properties of Limits with Trigonometry

12. §1.3: Properties of Limits with Trigonometry
Let 𝒃 and 𝒄 be real numbers, let 𝒏 be a positive integer, and let 𝒇 and 𝒈 be functions with the given limits: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 =𝑳 and 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 =𝑲
Scalar Multiple: 𝐥𝐢𝐦 𝒙→𝒄 𝒃𝒇 𝒙 = 𝒃 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙
Sum or Difference: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙
Product: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ∙ 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙
Quotient: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙
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§1.3: Properties of Limits with Trigonometry

13. §1.3: Properties of Limits with Trigonometry
Example 5
Find the following limit given 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) = 𝟕 𝟔 and 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) = 𝟓 𝟔 determine 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) + 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙)
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§1.3: Properties of Limits with Trigonometry

14. §1.3: Properties of Limits with Trigonometry
Example 6
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧𝟐(𝒙)
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§1.3: Properties of Limits with Trigonometry

15. §1.3: Properties of Limits with Trigonometry
Example 7
Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝒙 𝒆 𝒙
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§1.3: Properties of Limits with Trigonometry

16. §1.3: Properties of Limits with Trigonometry
Example 8
Given the graph of 𝒇 𝒙 , find 𝐥𝐢𝐦 𝒙→𝟏 (𝟓𝒇 𝒙 + 𝒇 𝟏 ) 𝐥𝐢𝐦 𝒙→𝟒 (𝒇 𝒙 ) 𝟐
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§1.3: Properties of Limits with Trigonometry

17. §1.3: Properties of Limits with Trigonometry
Your Turn
Solve the following limit given 𝐥𝐢𝐦 𝒙→−𝟏 𝒇 𝒙 =𝟑 𝒙 𝟐 −𝟐𝒙−𝟏 and 𝐥𝐢𝐦 𝒙→−𝟏 𝒈 𝒙 = 𝒙 𝟐 +𝟏 , determine 𝐥𝐢𝐦 𝒙→−𝟏 𝒉 𝒙 = 𝒇 𝒙 𝒈 𝒙
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§1.3: Properties of Limits with Trigonometry

18. “𝟎/𝟎” Limits AKA: Indeterminate Form
Always begin with direct substitution
Completely factor the problem
Simplify and/or Cancel by identifying a function 𝒈 that agrees with for all x except 𝒙 = 𝒄. Take the limit of 𝒈
Apply algebra rules
If necessary, Rationalize the numerator or denominator
Plug in 𝒙 of the function to get the limit
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§1.3: Properties of Limits with Trigonometry

19. §1.3: Properties of Limits with Trigonometry
Example 9
Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
What form is this?
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§1.3: Properties of Limits with Trigonometry

20. §1.3: Properties of Limits with Trigonometry
Example 9
Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
AS 𝒙 APPROACHES 4, 𝒇(𝒙) OR 𝒚 APPROACHES 8.
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§1.3: Properties of Limits with Trigonometry

21. §1.3: Properties of Limits with Trigonometry
Example 10
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐭𝐚𝐧 𝒙 𝐜𝐨𝐬 𝒙
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§1.3: Properties of Limits with Trigonometry

22. When in Algebra… NO RADICALS IN THE DENOMINATOR
You learned to:
NO RADICALS IN THE DENOMINATOR
FOR LIMITS, NO RADICALS IN THE NUMERATOR OR DENOMINATOR
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§1.3: Properties of Limits with Trigonometry

23. §1.3: Properties of Limits with Trigonometry
Example 11
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
What form is this?
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§1.3: Properties of Limits with Trigonometry

24. §1.3: Properties of Limits with Trigonometry
Example 11
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
NO NEED TO FOIL THE BOTTOM
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§1.3: Properties of Limits with Trigonometry

25. §1.3: Properties of Limits with Trigonometry
Example 11
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
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§1.3: Properties of Limits with Trigonometry

26. §1.3: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑
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§1.3: Properties of Limits with Trigonometry

27. §1.3: Properties of Limits with Trigonometry
Example 12
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
What form is this?
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§1.3: Properties of Limits with Trigonometry

28. §1.3: Properties of Limits with Trigonometry
Example 12
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3: Properties of Limits with Trigonometry

29. §1.3: Properties of Limits with Trigonometry
Example 12
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3: Properties of Limits with Trigonometry

30. §1.3: Properties of Limits with Trigonometry
Example 12
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3: Properties of Limits with Trigonometry

31. §1.3: Properties of Limits with Trigonometry
Example 13
Solve 𝐥𝐢𝐦 𝒙→𝒂 𝒙−𝒂 𝒙 𝟑 − 𝒂 𝟑
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§1.3: Properties of Limits with Trigonometry

32. §1.3: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙
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§1.3: Properties of Limits with Trigonometry

33. §1.3: Properties of Limits with Trigonometry
“Squeeze Theorem”
Also known as the “Sandwich theorem,” it is used to evaluate the limit of a function that can't be computed at a given point.
For a given interval containing point 𝒄, where 𝒇,  𝒈, and 𝒉 are three functions that are differentiable and 𝒈 𝒙 <𝒇 𝒙 <𝒉 𝒙 over the interval where 𝒇 𝒙 is the upper bound and 𝒉 𝒙 is the lower bound.
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§1.3: Properties of Limits with Trigonometry

34. §1.3: Properties of Limits with Trigonometry
“Squeeze Theorem”
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§1.3: Properties of Limits with Trigonometry

35. §1.3: Properties of Limits with Trigonometry
Example 14
Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) where 𝒄=𝟏 for 𝟑𝒙≤𝒈 𝒙 ≤ 𝒙 𝟑 +𝟐
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§1.3: Properties of Limits with Trigonometry

36. §1.3: Properties of Limits with Trigonometry
Example 14
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§1.3: Properties of Limits with Trigonometry

37. §1.3: Properties of Limits with Trigonometry
Example 15
Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝟒 𝒇(𝒙) for 𝟒𝒙−𝟗≤𝒇 𝒙 ≤ 𝒙 𝟐 −𝟒𝒙+𝟕 for which 𝒙≥𝟎
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§1.3: Properties of Limits with Trigonometry

38. §1.3: Properties of Limits with Trigonometry
Your Turn
Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) where 𝒄=𝟎 for 𝟗− 𝒙 𝟐 ≤𝒈 𝒙 ≤𝟗+ 𝒙 𝟐
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§1.3: Properties of Limits with Trigonometry

39. Special Trigonometric Limits
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 =𝟏
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝒙 𝒙 =𝟎
𝐥𝐢𝐦 𝒙→𝟎 𝟏+𝒙 𝟏/𝒙 =𝒆
When expressing 𝒙 in radians and not in degrees
Explains the “Squeeze” Theorem
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§1.3: Properties of Limits with Trigonometry

40. Why is the Limit of 𝐬𝐢𝐧 𝒙 𝒙 =𝟏 (as 𝒙 approaches to zero) ?
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§1.3: Properties of Limits with Trigonometry

41. Why is the Limit of 𝟏−𝐜𝐨𝐬 𝒙 𝒙 =𝟎, (as 𝒙 approaches to zero)?
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§1.3: Properties of Limits with Trigonometry

42. §1.3: Properties of Limits with Trigonometry
Example 16
Is there another way of rewriting 𝒕𝒂𝒏⁡(𝒙)?
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙
Split the fraction up so we can isolate and utilize a trigonometric limit
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§1.3: Properties of Limits with Trigonometry

43. §1.3: Properties of Limits with Trigonometry
Example 16
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙
Utilize the Product Property of Limits
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§1.3: Properties of Limits with Trigonometry

44. §1.3: Properties of Limits with Trigonometry
Example 17
Try to convert it to one of its trig limits.
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙
Try to get it where the sine trig function to cancel. Whatever is applied to the bottom, must be applied to the top.
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§1.3: Properties of Limits with Trigonometry

45. §1.3: Properties of Limits with Trigonometry
Example 17
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙
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§1.3: Properties of Limits with Trigonometry

46. §1.3: Properties of Limits with Trigonometry
Example 18
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙
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§1.3: Properties of Limits with Trigonometry

47. §1.3: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙
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§1.3: Properties of Limits with Trigonometry

48. §1.3: Properties of Limits with Trigonometry
Pattern?
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙 =𝟒
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙 = 𝟐 𝟑
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙 = 𝟓 𝟑
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟓𝒙 𝒙 = 𝟓
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟐𝐬𝐢𝐧 𝟑𝒙 𝟓𝒙 = 𝟔 𝟓
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§1.3: Properties of Limits with Trigonometry

49. §1.3: Properties of Limits with Trigonometry
Example 19
Split the fraction up so we can isolate and utilize a trigonometric limit
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙
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§1.3: Properties of Limits with Trigonometry

50. §1.3: Properties of Limits with Trigonometry
Example 19
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙
cos(0) = 1
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§1.3: Properties of Limits with Trigonometry

51. §1.3: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟑−𝟑 𝐜𝐨𝐬 𝒙 𝒙
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§1.3: Properties of Limits with Trigonometry

52. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐬𝐢𝐧 𝒙 𝒙 (A) 𝟎 (B) –𝝅/𝟐 (C) 𝟐 𝟐 /𝝅 (D) 𝟐/𝝅
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§1.3: Properties of Limits with Trigonometry

53. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐬𝐢𝐧 𝒙 𝒙
Vocabulary
Connections and Process
Answer and Justifications
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§1.3: Properties of Limits with Trigonometry

54. §1.3: Properties of Limits with Trigonometry
Assignment
Page EOO, 23, 25, odd, odd (omit 45), odd, 89
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§1.3: Properties of Limits with Trigonometry