1. 2.2 The Concept of Limit Wed Sept 16 Do Now Sketch the following functions. Describe them.

2. HW: p.64 #1 5 7 1) a- 11.025 m b- 22.05 m/s c- ~19.6 m/s 5) 0.3 m/s 7) a- dollars/year b- [0, 0.5]: 7.8461; [0, 1]: 8 c- ~ $8/year

3. Concept of a Limit Take note of the two functions. Both functions are undefined at x = 2 Let’s take a look at each function for x close to 2

4. F(x) and g(x) X F(x)G(x) 1.913.93.9 1.99103.993.99 1.9991003.9993.999 1.999910003.99993.9999 1.99999100003.99993.99999 1.99999910000043.999999

5. Limits We consider the limit of a function as the value the function approaches as it gets closer to a certain value. In the table, we approached x = 2 from the left side. We denote this as

6. Limits Cont’d Repeat the same process from the right side

7. One-sided Limits Limits from the left or right side of a function are called one-sided limits If two one-sided limits of f(x) are the same, they comprise the limit of f(x) Important: A limit exists if and only if both one-sided limits exist and are equal.

8. Limits and Graphs For now, we’ll be using graphs and tables to see if limits exist or not A graphing calculator helps when looking at functions to determine where the limits exist

9. Exs in book

10. Canceling Factors If a function has identical factors in the numerator and denominator, they can be cancelled before finding the limit Canceling factors will not affect the limit of the function

11. Piecewise functions When looking at piecewise functions, it is often important to use one-sided limits to determine if a limit exists. Absolute value functions are included in this idea

12. Closure What is a one-sided limit? What is the notation involved with limits? How do we know if a limit exists? What must be true? Homework: pp 74-75 #1, 3, 5, 6, 38, 47, 53

13. 2.2 Limits using Graphs/Tables Thurs Sep 17 Do Now Let f(x) = x + 3g(x) = 4 / (x - 3) 1) Find 2) Find

14. HW Review: p.74 #1 3 5 6 38 47 53 1) 3/247) c = 2 (inf, inf) 3) 3/5 c= 4 (- inf, 10) 5) 1.5v.a. x = 2 6) 1.553) c = 1 (3, 3) 38) c = 1 (3, 1) DNEc = 3 (-inf, 4) c = 2 (2, 1) DNEc = 5 (2, -3) c = 4 (2, 2) existsc = 6 (inf, inf)

15. Review A limit is the y-value a function approaches as x gets close to something It does NOT matter what the function is AT that point…only what it seems to approach!

16. How to compute limits? For now, we can use either a graph or a table to determine a function’s limit Use tables when it is difficult to determine where a graph is approaching (not whole numbers)

17. Practice Worksheet 2.2 Quiz tomorrow –Limits Graphs Table

18. Closure Graphand find HW: Finish selected problems on worksheet Quiz tomorrow

19. 2.2 Quiz Fri Sept 18 Do Now Find the left and right hand limits of

20. HW Review: worksheet p.110- 111 #1-12

21. 2.2 Quiz