1. Homework Homework Assignment #3 Read Section 2.4 Page 91, Exercises: 1 – 33 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 82 Evaluate the limits using the Limit Laws and the following where c and k are constants. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 82 Evaluate the limit assuming: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 82 29. Can the Quotient Law be used to evaluate : ? Explain. If the limit is rewritten as: using the Product Law, we obtain which does not exist, resulting in a nonexistent limit. Using the Quotient Law, direct substitution yields an indeterminate form, resulting in another nonexistent limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 82 33. Use the Limit Laws and the result to show that for all whole numbers. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 2: Limits Section 2.4: Limits and Continuity

12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If the graph of a function may be drawn without lifting the pencil from the page, the graph and thus the function are said to be continuous, such as at x = c in Figure 1. A break in the graph, such as at x = c in Figure 2, is called a discontinuity.

13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The discussion on the previous slide leads to the following definition: If a continuous function is defined on [a, b], and c is any point on (a, b) then the following conditions exist:

14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Three conditions must hold for a function to be continuous: Figures 3 and 4 are graphs of continuous functions.

15. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Some functions contain discontinuities that can be removed by redefining the piecewise function, such as defining f (2) = 5 instead of f (2) = 10 for the function graphed in Figure 5. Such discontinuities are called removable discontinuities.

16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A jump discontinuity is a discontinuity for which the first condition of continuity is not met, that is: If either, the function is said to be one-sided continuous as defined below:

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A jump discontinuity may be one-sided continuous as in Figure 6 (A) or neither right– nor left–continuous as in 6(B). A function with a jump discontinuity may not be both right– and left–continuous at the jump discontinuity, as it would no longer pass the vertical line test, meaning it is no longer a function.

18. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company An infinite discontinuity is one in which one or both one- sided limits are infinite. Since infinity is a concept rather than an actual number, f (2) is not defined in Figure 8 (A) or 8 (B). Some texts state that the limit at x = 2 in Figure 8 (A) does not exist, although ours does not so state.

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 9 illustrates what some refer to as an oscillating discontinuity. In this case, neither the right– nor left– sided limits exist at x = 0, so neither does the limit.

20. Example, Page 91 2. Find the points of discontinuity and state whether f (x) is left- or right-continuous, or neither at these points. At which point does f (x) have a removable discontinuity? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Functions “built” of functions known to be continuous are also continuous in accordance with Theorem 1. Theorem 1 is proven as follows:

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As shown in the proof of Theorem 2, all polynomials are continuous.

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 10 shows the graphs of some basic functions that are continuous on their domains. This is true of most basic functions.

24. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Theorem 3 formally states the continuity of some basic functions. Since inverse functions are reflections of the parent function about the function y = x, the inverses of continuous functions are also continuous on their domains, as stated in Theorem 4.

25. Example, Page 91 Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

26. Example, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Example, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

28. Example, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Just as products and quotients of continuous functions are continuous on their domains, compositions of continuous functions are also continuous as stated in Theorem 5.

30. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company When a function f (x) is known to be continuous, then we may state. This method of evaluating a limit is sometimes called the substitution method as the value of c is substituted for x in the function to obtain the limit. The substitution method can not be used on the greatest integer function shown in Figure 12 as it is not everywhere continuous.

31. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Sometimes continuous functions are used to model physical quantities that on a large-scale may mimic a continuous function, but on a small scale these quantities are incremental and thus discontinuous. Figure 13 shows two such functions.

32. Homework Homework Assignment #4 Read Section 2.5 Page 91, Exercises: 1 – 33 (EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company