1. Numerical Sequences

2. Why Sequences? There are six animations about limits to show the sequence in the domain and range. Problems displaying the data. It has to be fixed Intuition about Infinitely close and direction

3. Why do we need to approach to the limit point systematically?

4. The behavior of the function g(x)=1/x was previously discussed for values of x "close to zero", for values of x "very large but positive", and for values of x "very large but negative".

5. Support from Tables Verify these tables with the calculator

6. Challenge

7. The graph of a function f(x) is given but you don’t know anything else about the function other than its graph and values of the function at points in the domain which are "close" to zero. Use the graph of the function y=f(x) to make conjectures about the following values Are these conjectures supported by the tables?

8. The Graph of the same function in two Different Windows Discuss the limits below on both windows. Are they the same? Should they be the same?

9. Definitionof Numerical Sequences Listing – Finite – Infinite Formulas (graph as listing and as functions) a. b. c. d.

10. Basic Functions vs.. Basic Sequences Show graphing sequences Indicate how to start always at zero Sequence as a listing vs. sequence as a function

11. Key types of sequences Diverging to infinity Converging to zero

12. Sequences Diverging to Infinity Number Positive Infinitely Large “∞” Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER Slider Normally a LARGE NUMBER

13. EXAMPLE A=800 When N=29 First term passing slider What is N when A=10 28 ? Show in calculator

14. Exercise For the sequence below find the number of terms less than or equal to each of the given values of A. Represent the results geometrically in one and two dimensions. For each A determine the value N satisfying the condition

15. Any number within ε from 0 approximates 0 with accuracy less than ε Positive Sequences Converging to Zero Numbers Positive Infinitely Small “0 +” SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero SLIDER ε (small) distance from zero

16. Any term of the sequence here represents 0 accurate to three decimal places EXAMPLE When N=1001 first term passing SLIDER. The term is a 1001= 1/1001 a 1001 =0.000999 When N=1001 first term passing SLIDER. The term is a 1001= 1/1001 a 1001 =0.000999

17. When N=1001 first term passing SLIDER. The term is a 1001 =1/1001 a 1001 =0.000999 When N=1001 first term passing SLIDER. The term is a 1001 =1/1001 a 1001 =0.000999 Show in calculator What is N when ?

18. EXERCISE For the sequence find the number of terms less than or equal to the given values of. Start by construction the table of values of the sequence, and represent the results on the number line/graph. Finally support your conclusions algebraically.

19. Relating Negative Infinitely Large and Negative Infinitely Small Numbers Positively Large Numbers Positively Small Numbers Yield

20. Negative Infinitely large numbers are the opposite of Positive Large Numbers Negative Infinitely Small Numbers are the opposite of Positive Infinitely Small Numbers

21. Relating Positive Infinitely Large and Positive Infinitely Small Numbers Negative Large Numbers Negative Small Numbers Yield

22. Converging to Any Number Other Than Zero Generate three examples that represent each of the following expressions “7 + ”, “7 - ”, “7”

23. Key Observation “0 +” are always positive “0 -” are always negative “0” could be either “0 +” or “0 -” “7 + ”, “7 - ”, “7” represent always positive numbers (infinitely close to 7) “a” means numbers infinitely close to a, a any real number

24. Verification Time Provide four examples that represent each of the following expressions – “∞” – “0” – “0.2 +” – “_ 0.002 -”

25. Operations with “ “ Operation between sequences are point-wise or term by term

26. Notation c ×”∞"=“∞", when c is positive c ×”∞"=“-∞", when c is negative Represents any sequence converging to zero from the right added to a sequence diverging to infinity

27. Addition

28. Product

29. Quotient

30. Trouble-Makers Each situation has to be analyzed. No general rules as in the other cases

31. Dominance of Sequences The sequence a n DOMINATES the sequence b n as n goes to infinity means that

32. Exercise Show that the sequence on the left column dominates the sequence on the second column

33. Exercise 4 Workbook 4.3 Two exercises assigned to each group.

34. Transformation of sequences under Functions indicates the behavior of the heights of the function f(x) for values of x in the domain infinitely close to a from the right. The values of x could be "any sequence” converging to a from the right.

35. Consider the function

36. Exercise 2