2. Express the repeating decimal 0.5757... as the ratio of two integers without your calculator. Warm-Up

3. Express the repeating decimal 0.5757... as the ratio of two integers without your calculator. Warm-=Up x = 0.57 100x = 57.57 99x = 57 x = 57/99

4. What is Calculus? There are only 3 main concepts in calculus. 1)The Limit 2) The Derivative 3)The Integral

5. What is Calculus? There are only 3 main concepts in calculus. 1)The Limit 2) The Derivative 3)The Integral 4)You will need a graphing calculator.

6. 1-2:Finding Limits Graphically and Numerically Objectives: Understand the concept of a limit Calculate limits

7. Important Ideas Limits are what make calculus different from algebra and trigonometry Limits are fundamental to the study of calculus Limits are related to rate of change Rate of change is important in engineering & technology

8. Analysis Slope is a rate of change Rate of change is constant at every value on a linear f(x) m=2 f(x) x m=3 m=2 m=1 m=-1

9. Analysis f(x) x Rate of change is different at every value on a non-linear f(x) Rate of change is the slope of the tangent line at a point

10. Important Ideas The slope of a secant line is an average rate of change The slope of a tangent line is an instantaneous rate of change at a point

11. We know how to calculate average rate of change Analysis The tangent line problem… Go to Sketchpad We don’t know how to calculate instantaneous rate of change,therefore,

12. Warm-Up-You need a graphing calculator. I’m using a TI-84. Put your signature pages in the box

13. Important Idea Instantaneous Rate of change is different at every point on f(x) f(x) x Limits are used to calculate the slopes of the tangents

14. Example 1. Graph: 2. Trace to x =2. 3. Zoom in at least 4 times. 4. Describe the graph.

15. Example Consider What happens at x =1? x.75.9.99.999 f(x) Let x get close to 1 from the left:

16. Try This Consider x 1.251.11.011.001 f(x) Let x get close to 1 from the right:

17. Try This What number does f(x) approach as x approaches 1 from the left and from the right?

18. Try This Graph and on the same axes. What is the difference between these graphs?

19. Why is there a “hole” in the graph at x =1? Analysis

20. Example Consider for and for x =1 =?

21. Try This Find: if

22. Important Idea The existence or non- existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.

23. Important Idea What matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.

24. Try This Find: f(0) is undefined; 2 is the limit 2

25. Find: Try This f(0) is defined; 2 is the limit 2 1

26. Warm-Up

27. Try This Find the limit of f(x) as x approaches 3 where f is defined by:

28. Example Graph and find the limit (if it exists):

29. Important Idea Some limits do not exist. If f(x) approaches as x approaches c, we say that the limit does not exist at c or, sometimes we say the AP Exam says the limit approaches infinity at c.

30. Example Find the limit if it exists:

31. Important Idea But… Does not exist

32. Definition If a function has a limit, the limit from the right must equal the limit from the left.

33. Example 1.Graph using a friendly window: 2. Zoom at x =0 3. Wassup at x =0?

34. Important Idea If f(x) bounces from one value to another (oscillates) as x approachs c, the limit of f(x) does not exist at c :

35. Lesson Close Name 3 ways a limit may fail to exist.

36. Assignment Page 54 Problems 1 - 7 odd, 8 – 24 all In class, we will not cover the formal definition of a limit, sometimes called epsilon-delta definition. I’ll talk about it in NMSI tutoring.