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

2. “ 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
Plug in 𝒙 of the function to get the limit
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§1.3A: Properties of Limits with Trigonometry

3. §1.3A: Properties of Limits with Trigonometry
Example 1
Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
What form is this?
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§1.3A: Properties of Limits with Trigonometry

4. §1.3A: Properties of Limits with Trigonometry
Example 1
Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
AS 𝒙 APPROACHES 4, 𝒇(𝒙) OR 𝒚 APPROACHES 8.
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§1.3A: Properties of Limits with Trigonometry

5. §1.3A: Properties of Limits with Trigonometry
Example 1 (Calculator)
Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
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§1.3A: Properties of Limits with Trigonometry

6. §1.3A: Properties of Limits with Trigonometry
Example 2
Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝟐𝒙 𝟐 −𝒙−𝟑 𝒙 𝟐 −𝟐𝒙−𝟑
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§1.3A: Properties of Limits with Trigonometry

7. §1.3A: Properties of Limits with Trigonometry
Example 3
Solve 𝐥𝐢𝐦 𝒙→𝒂 𝒙−𝒂 𝒙 𝟑 − 𝒂 𝟑
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§1.3A: Properties of Limits with Trigonometry

8. §1.3A: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒕→−𝟏 𝒕 𝟑 −𝒕 𝒕 𝟐 −𝟏
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§1.3A: Properties of Limits with Trigonometry

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

10. §1.3A: Properties of Limits with Trigonometry
Example 4
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
What form is this?
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§1.3A: Properties of Limits with Trigonometry

11. §1.3A: Properties of Limits with Trigonometry
Example 4
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
NO NEED TO FOIL THE BOTTOM
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§1.3A: Properties of Limits with Trigonometry

12. §1.3A: Properties of Limits with Trigonometry
Example 4
Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗
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§1.3A: Properties of Limits with Trigonometry

13. §1.3A: Properties of Limits with Trigonometry
Example 5
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟏 −𝟏 𝒙
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§1.3A: Properties of Limits with Trigonometry

14. §1.3A: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑 . Hint: Don’t combine like terms to the denominator, too early
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§1.3A: Properties of Limits with Trigonometry

15. §1.3A: Properties of Limits with Trigonometry
Example 6
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
What form is this?
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§1.3A: Properties of Limits with Trigonometry

16. §1.3A: Properties of Limits with Trigonometry
Example 6
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3A: Properties of Limits with Trigonometry

17. §1.3A: Properties of Limits with Trigonometry
Example 6
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3A: Properties of Limits with Trigonometry

18. §1.3A: Properties of Limits with Trigonometry
Example 6
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙
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§1.3A: Properties of Limits with Trigonometry

19. §1.3A: Properties of Limits with Trigonometry
Example 7
Evaluate 𝐥𝐢𝐦 𝒙→𝟏 𝟏 𝒙+𝟏 − 𝟏 𝟐 𝒙−𝟏
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§1.3A: Properties of Limits with Trigonometry

20. §1.3A: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙
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§1.3A: Properties of Limits with Trigonometry

21. §1.3A: 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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§1.3A: Properties of Limits with Trigonometry

22. §1.3A: Properties of Limits with Trigonometry
“Squeeze Theorem”
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§1.3A: Properties of Limits with Trigonometry

23. §1.3A: Properties of Limits with Trigonometry
Example 8
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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§1.3A: Properties of Limits with Trigonometry

24. §1.3A: Properties of Limits with Trigonometry
Example 8
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§1.3A: Properties of Limits with Trigonometry

25. §1.3A: Properties of Limits with Trigonometry
Example 9
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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§1.3A: Properties of Limits with Trigonometry

26. §1.3A: 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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§1.3A: Properties of Limits with Trigonometry

27. Special Trigonometric Limits
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 =𝟏
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝒙 𝒙 =𝟎
𝐥𝐢𝐦 𝒙→𝟎 𝟏+𝟏/𝒙 𝒙 =𝒆
When expressing 𝒙 in radians and not in degrees
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§1.3A: Properties of Limits with Trigonometry

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

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

30. §1.3A: Properties of Limits with Trigonometry
Example 10
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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§1.3A: Properties of Limits with Trigonometry

31. §1.3A: Properties of Limits with Trigonometry
Example 10
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙
Utilize the Product Property of Limits
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§1.3A: Properties of Limits with Trigonometry

32. §1.3A: Properties of Limits with Trigonometry
Example 11
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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§1.3A: Properties of Limits with Trigonometry

33. §1.3A: Properties of Limits with Trigonometry
Example 11
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙
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§1.3A: Properties of Limits with Trigonometry

34. §1.3A: Properties of Limits with Trigonometry
Example 12
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙
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§1.3A: Properties of Limits with Trigonometry

35. §1.3A: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙
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§1.3A: Properties of Limits with Trigonometry

36. §1.3A: 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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§1.3A: Properties of Limits with Trigonometry

37. §1.3A: Properties of Limits with Trigonometry
Example 13
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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§1.3A: Properties of Limits with Trigonometry

38. §1.3A: Properties of Limits with Trigonometry
Example 13
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙
cos(0) = 1
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§1.3A: Properties of Limits with Trigonometry

39. §1.3A: Properties of Limits with Trigonometry
Your Turn
Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟑−𝟑 𝐜𝐨𝐬 𝒙 𝒙
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§1.3A: Properties of Limits with Trigonometry

40. AP Multiple Choice Practice Question 1 (non-calculator)
If 𝒂≠𝟎, then determine 𝐥𝐢𝐦 𝒙→𝒂 𝒙 𝟐 − 𝒂 𝟐 𝒙 𝟒 − 𝒂 𝟒 (A) 𝟏 𝒂 𝟐 (B) 𝟏 𝟐𝒂 𝟐 (C) 𝟏 𝟔𝒂 𝟐 (D) 𝟎
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§1.3A: Properties of Limits with Trigonometry

41. AP Multiple Choice Practice Question 1 (non-calculator)
If 𝒂≠𝟎, then determine 𝐥𝐢𝐦 𝒙→𝒂 𝒙 𝟐 − 𝒂 𝟐 𝒙 𝟒 − 𝒂 𝟒
Vocabulary
Connections and Process
Answer and Justifications
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§1.3A: Properties of Limits with Trigonometry

42. §1.3A: Properties of Limits with Trigonometry
Assignment
Page odd, odd, odd, 89
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§1.3A: Properties of Limits with Trigonometry