1. Calculus Limits Lesson 2

2. Bell Activity A. Use your calculator graph to find: 1. 2. 3. 4.

3. B. Without Calculator, Find 1. f(2) if f(x) = 3x – 2 2. f(-3) if f(x) = 2x + 5 3. f(4) if f(x) = 4 – x/2 4. f(2) if f(x) = Notice: In #4, both the numerator and denominator are 0. This is called an indeterminate form and thus cannot be evaluated.

4. The limit may be estimated or found by creating a table of values very close to the approaching x value. X 1.75 1.9 1.99 2 2.001 2.01 2.1 Y.75.9.99 1.001 1.01 1.1 =1

5. X 2.8 2.9 2.99 3 3.001 3.01 3.1 Y *** What do you think allows this function to have the limit as x approaches 3 and f(3) to be the same whereas the first problem does not? It would seem from this example that at times a limit of a function as x approaches a particular # can simply be obtained by finding f(that #). =1.6.8.981 1.002 1.021.2

6. = 1

7. Which of these can be substituted directly to find the limit? 1. 2. 3. Even though # 3’s limit can’t be found by substituting directly we can still get the limit from a graph or a table. =11 =3 =2

8. Let’s go back to a problem where the substitution would not work to find the limit. Perhaps we could modify the problem before we substituted. =2

9. Modify these and find the limit by substituting: Let’s verify these with the calculator graph.

10. Find the limit: Let’s verify with the calculator graph.

11. Find the limit: Let’s verify with the calculator graph.

12. There are other times when we cannot find the limit by either substituting or factoring. In this case, Rationalize the Numerator

13. And, of course, there are problems which merely need to be simplified before substituting.

14. Find the limit: Let’s verify with the calculator graph.

15. Find the limit: Let’s verify with the calculator graph.

16. Find the limit: Let’s verify with the calculator graph.

17. Slide 2- 17 Properties of Limits

18. Slide 2- 18 Properties of Limits continued Product Rule: Constant Multiple Rule:

19. Properties of Limits continued

20. Example Properties of Limits

21. Assignment Text p. 67, # 1 – 43 odds