1. Lesson 11.2 Techniques for Evaluating Limits
Essential Question: How do you evaluate limits that cannot be solved through use of direct substitution?

2. Before we start…
Evaluate the limit. lim 𝑥→−3 𝑥 2 +𝑥−6 𝑥+3

3. What happens when you get 0/0?
If you use direct substitution and get 0 0 , then you have no real answer because you have the indeterminate form. You must use a dividing out technique or rationalizing technique before you can determine that the limit does not exist.

4. Dividing Out Technique
Factor the numerator
Factor the denominator
Cancel
Use direct substitution to evaluate

5. Evaluate the limit. lim 𝑥→−3 𝑥 2 +𝑥−6 𝑥+3

6. Evaluate the limit. lim 𝑥→2 𝑥 2 +2𝑥−8 𝑥−2

7. Evaluate the limit. lim 𝑥→1 𝑥−1 𝑥 3 − 𝑥 2 +𝑥−1

8. Evaluate the limit. lim 𝑥→2 𝑥−2 𝑥 3 − 2𝑥 2 +2𝑥−4

9. Evaluate the limit. lim 𝑥→0 𝑥+2 2 −4 𝑥

10. Evaluate the limit. lim 𝑥→0 𝑥+3 2 −9 𝑥

11. Rationalizing Technique
Multiply by the conjugate
Simplify
Use direct substitution to evaluate

12. Evaluate the limit. lim 𝑥→0 𝑥+1 −1 𝑥

13. Evaluate the limit. lim 𝑥→0 𝑥+9 −3 𝑥

14. Evaluate the limit. lim 𝑥→3 𝑥+1 −2 𝑥−3

15. Evaluate the limit. lim 𝑥→0 1 𝑥+4 − 1 4 𝑥

16. Evaluate the limit. lim 𝑥→0 1 𝑥−8 + 1 8 𝑥

17. For the function given by 𝑓 𝑥 = 𝑥 2 −1, find lim ℎ→0 𝑓 3+ℎ −𝑓 3 ℎ

18. For the function given by 𝑓 𝑥 = 2𝑥 2 +1, find lim ℎ→0 𝑓 2+ℎ −𝑓 2 ℎ

19. How do you evaluate limits that cannot be solved through use of direct substitution?

20. Ticket Out the Door
Evaluate the limit. lim 𝑥→2 𝑥 2 −𝑥−2 𝑥−2