1. Evaluating Limits Analytically
Section 1.3
Calculus AP/Dual, Revised ©2017
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§1.3: Evaluating Limits Analytically

2. When does a Limit Fail to Exist?
Left side does not equal to the Right side
Graph oscillates
Limit has an unvalued behavior (such as ∞)
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§1.3: Evaluating Limits Analytically

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: Evaluating Limits Analytically

4. §1.3: Evaluating Limits Analytically
Example 1
Evaluate 𝐥𝐢𝐦 𝒙→𝟑 𝟐𝒙+𝟓
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§1.3: Evaluating Limits Analytically

5. §1.3: Evaluating Limits Analytically
Example 2
Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝟏𝟎
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§1.3: Evaluating Limits Analytically

6. §1.3: Evaluating Limits Analytically
Example 3
Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟐 −𝟏 𝒙 𝟑
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§1.3: Evaluating Limits Analytically

7. §1.3: Evaluating Limits Analytically
Example 4
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙
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§1.3: Evaluating Limits Analytically

8. §1.3: Evaluating Limits Analytically
Example 5
Find the following limits given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 , determine 𝐥𝐢𝐦 𝒙→−𝟓 𝒇 𝒙
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§1.3: Evaluating Limits Analytically

9. §1.3: Evaluating Limits Analytically
Example 6
Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙−𝟐
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§1.3: Evaluating Limits Analytically

10. §1.3: Evaluating Limits Analytically
Your Turn
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝟐𝒙−𝟏 𝟑
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§1.3: Evaluating Limits Analytically

11. 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: Evaluating Limits Analytically

12. §1.3: Evaluating Limits Analytically
Review Example A
If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒙
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§1.3: Evaluating Limits Analytically

13. §1.3: Evaluating Limits Analytically
Review Example B
If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒈 𝒇 𝒙
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§1.3: Evaluating Limits Analytically

14. §1.3: Evaluating Limits Analytically
Example 7
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: Evaluating Limits Analytically

15. §1.3: Evaluating Limits Analytically
Example 8
Find the following limit given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 −𝟏, determine 𝐥𝐢𝐦 𝒙→−𝟏 𝒇 𝒈 𝒙
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§1.3: Evaluating Limits Analytically

16. §1.3: Evaluating Limits Analytically
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: Evaluating Limits Analytically

17. §1.3: Evaluating Limits Analytically
Properties of Limits
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: Evaluating Limits Analytically

18. §1.3: Evaluating Limits Analytically
Example 9
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) +𝒈(𝒙)]
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§1.3: Evaluating Limits Analytically

19. §1.3: Evaluating Limits Analytically
Example 10
Find 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 𝒇 𝒙 = 𝒇 𝒙 𝒈 𝒙
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§1.3: Evaluating Limits Analytically

20. §1.3: Evaluating Limits Analytically
Example 11
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: Evaluating Limits Analytically

21. §1.3: Evaluating Limits Analytically
Your Turn
Find the following limit given  𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 = 𝟑 𝟐 and 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 = 𝟏 𝟐 , determine 𝐥𝐢𝐦 𝒙→𝒄 𝟒 𝒇 𝒙
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§1.3: Evaluating Limits Analytically

22. 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: Evaluating Limits Analytically

23. 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: Evaluating Limits Analytically

24. §1.3: Evaluating Limits Analytically
Assignment
Page EOO, 23, 25, odd (omit 45)
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§1.3: Evaluating Limits Analytically