Math 142, Chapter 3 Spring 2011 Zarestky Chapter 3 Reading Notes In this chapter, we will learn about limits of functions and then use limits to transition into derivatives, the first real concept of calculus. The overarching idea here is infinity. With limits, you will approach infinity, or get infinitely close to a value, but you may never actually reach infinity or the value. It is a new way of looking at functions and their behavior. Section 3.1: Introduction to Limits By the end of this section, you should be able to: • Estimate a limit at a point from the graph of a function. • Determine the values of x for which a limit does not exist given a graph of a function. • Evaluate limits at a point algebraically. Read the section Functions and Graphs: Brief Review and Limits: A Graphical Approach, through Example 3. (About 3 pages. Examples 1 and 2 should be review. Example 3 will be new, but there isn’t much actual work to do. Focus on understanding.) These first few pages should align with your prior experience with functions. Pay special attention to the note in the Definition of a Limit blue box. That will be particularly important later. Make notes about the mathematical notation of limits. Continue on to read the paragraphs about One-Sided Limits and Theorem 1. Make notes about the notation and meaning of one-sided limits and the existence of a limit. Work Example 4. There isn’t actually much work to do, but make certain you think about and understand the book’s solution. Try the Matched Problem 4 on your own. What questions do you have?Math 142, Chapter 3 Spring 2011 Zarestky Make notes about Theorem 2, Properties of Limits. It is a lot of math-y formulas, but the bottom line is that everything works the way we’d intuitively like it to. Work Example 5. Compare your answers to the book’s solution. What questions do you have? Continue reading. Work Examples 6, 7, and 8. In examples 7 and 8, watch out for the special left and right notation. What questions do you have? Read and make notes about the Indeterminate Form and the Limit of a Quotient.Math 142, Chapter 3 Spring 2011 Zarestky Examples 9-11 deal with the Limits of Difference Quotients. These special types of limits are going to be very important when we get to derivatives. In the meantime, the toughest part is the algebra involved. Be careful! Work Example 9. Compare your answers to the book’s solution. What questions do you have? Work Example 10. Compare your answers to the book’s solution. What questions do you have? Work Example 11. Compare your answers to the book’s solution. What questions do you have? Check your understanding of this section by working a few problems from the exercises on your own. Focus on 1-15(odd), 17-20, 21-57(odd), 65-68, 71, 73, 75. What questions do you still have?Math 142, Chapter 3 Spring 2011 Zarestky Section 3.2: Infinite Limits and Limits at Infinity By the end of this section, you should be able to: • Estimate a limit at a point and a limit at infinity from the graph of a function. • Determine the values of x for which a limit does not exist given a graph of a function. • Evaluate limits at a point algebraically. • Evaluate limits at infinity for rational functions. In this section we will look at limits for which x ! ±" and limits at vertical asymptotes. Read the pages on Infinite Limits and Locating Vertical Asymptotes. Make notes about the definition and the special case of rational functions. (What is a rational function? If you don’t know, go look it up. Section 2.4 in your book.) How can you recognize when a rational function has a vertical asymptote? Work Example 1. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 3 Spring 2011 Zarestky Work Example 2. Compare your answers to the book’s solution. What questions do you have? Read the paragraphs on Limits at Infinity. Make notes about how to read a limit at infinity from a graph, horizontal asymptotes. Verify or Theorem 2 by graphing the functions from parts 1-4 on your graphing calculator by choosing specific values of p and k.Math 142, Chapter 3 Spring 2011 Zarestky Work Example 3. Compare your answers to the book’s solution. What questions do you have? Read Theorem 3 and the corresponding text. Be prepared to use the terminology leading term and end behavior. If you need to review polynomials, refer to section 2.4. Work Example 4. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 3 Spring 2011 Zarestky Read the paragraphs on Finding Horizontal Asymptotes. Make notes about Theorem 4. Be prepared to articulate the three cases in your own words. Work Example 5. Compare your answers to the book’s solution. What questions do you have? Work Example 6. Compare your answers to the book’s solution. What questions do you have? Check your understanding of this section by working a few problems from the exercises on your own. Focus on 1-8, 9-15(odd), 21-51(odd), 61-75(odd). What questions do you still have?Math 142, Chapter 3 Spring 2011 Zarestky Section 3.3: Continuity By the end of this section, you should be able to: • Identify the intervals on which a function is continuous given its graph or its equation. • Locate points of discontinuity for a piece-wise function. By now you’ve seen that there are features of a graph that are analyzed by limits, such as asymptotes or “jumps.” Those features affect the continuity of a function. Read the paragraphs on Continuity. Make notes about reading a graph and the definition of continuity. What are some ways you can recognize that a function is not continuous? Work Example 1. Compare your answers to the book’s solution. What questions do you have? Work Example 2. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 3 Spring 2011 Zarestky Read the paragraphs on Continuity Properties. Make notes about continuity properties for specific functions. Work Example 3. Compare your answers to the book’s solution. What questions do you have? Read the paragraphs on Solving Inequalities Using Continuity Properties. Make notes about solving inequalities and