1. Evaluating Limits Analytically
Section 12.2
Precalculus PreAP/Dual, Revised ©2017
4/20/2019 8:57 AM
12.2: Evaluating Limits

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
4/20/2019 8:57 AM
12.2: Evaluating Limits

3. Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 What form is this?
12.2: Evaluating Limits

4. Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒
AS X APPROACHES 4, f(x) OR Y APPROACHES 8.
4/20/2019 8:57 AM
12.2: Evaluating Limits

5. Example 1 (Calculator) Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 4/20/2019 8:57 AM
12.2: Evaluating Limits

6. Your Turn Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙 𝟐 +𝒙−𝟔 𝒙+𝟑 4/20/2019 8:57 AM
12.2: Evaluating Limits

7. Example 2 Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝟐𝒙 𝟐 −𝒙−𝟑 𝒙 𝟐 −𝟐𝒙−𝟑 4/20/2019 8:57 AM
12.2: Evaluating Limits

8. Example 3 Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐭𝐚𝐧 𝒙 𝐜𝐨𝐬 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

9. Your Turn Solve 𝐥𝐢𝐦 𝒕→−𝟏 𝒕 𝟑 −𝒕 𝒕 𝟐 −𝟏 4/20/2019 8:57 AM
12.2: Evaluating Limits

10. When in Algebra… NO RADICALS IN THE DENOMINATOR
You learned to:
NO RADICALS IN THE DENOMINATOR
FOR LIMITS, NO RADICALS IN THE NUMERATOR
4/20/2019 8:57 AM
12.2: Evaluating Limits

11. Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 What form is this? 4/20/2019 8:57 AM
12.2: Evaluating Limits

12. Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 NO NEED TO FOIL THE BOTTOM
12.2: Evaluating Limits

13. Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 4/20/2019 8:57 AM
12.2: Evaluating Limits

14. Example 5 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟏 −𝟏 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

15. Your Turn
Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑 . Hint: Don’t combine like terms to the denominator, too early
4/20/2019 8:57 AM
12.2: Evaluating Limits

16. Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 What form is this?
12.2: Evaluating Limits

17. Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

18. Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

19. Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

20. Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙 4/20/2019 8:57 AM
12.2: Evaluating Limits

21. Assignment Page 833 5-7, 9-21 odd 4/20/2019 8:57 AM
12.2: Evaluating Limits