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Unformatted text preview: 16 Chapter 1: Limits and Continuity Summary: This chapter explores the behavior of functions as they approach certain xvalues. First, some graphical and numerical methods are used to try to ascertain the behavior of a functions output values near a particular input value. Later, the chapter introduces some algebraic techniques for finding exact answers to this question. This idea is made more precise using ε and δ to help represent the tolerances in the outputs and inputs respectively while identifying how a function behaves. Generally, if a function approaches a consistent output value as the input values become closer to one value (such as x = a ), then a limit of the function values is said to exist (i.e., lim x → a f ( x ) = L ). Naturally, such limits can be discussed while input values approach x = a from the positive side, the negative side, or both. This directional idea is described accordingly as onesided or twosided limits. Limits become an even more useful tool when they are used to describe the no tion of a function being continuous. A function that is continuous can be said to have the property of lim x → a f ( x ) = f ( a ) which implies a limiting value, a func tion definition, and that the two values are equal. This formalizes the idea from the last chapter of polynomials having unbroken curves. Now polynomials can be described as being continuous everywhere. As it turns out, the trigonometric func tions sin x and cos x are continuous everywhere, which greatly aids in studying the behavior of all six trigonometric ratios. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Describe the behavior of a function using limits (§1 . 1 , 1 . 4). 2. Evaluate the limiting behavior of a function graphically, numerically or al gebraically (§1 . 1 , 1 . 2). 3. Discuss the relationship between vertical asymptotes and infinite limits (§1 . 1). 4. Discuss the relationship between horizontal asymptotes and the end behav ior of a function (§1 . 3). 5. Describe the connection between limits and continuity (§1 . 5). 17 18 6. Use the continuity of a function to evaluate the limit of a function at a point (§1 . 2 , 1 . 5). 7. Determine on what intervals a function is continuous (§1 . 5). 8. Describe the continuity of an inverse function (§1 . 6). 9. Use the Squeezing Theorem to evaluate limits (§1 . 6). 10. Use the IntermediateValue Theorem to approximate roots of a function (§1 . 5). 1.1 Limits (An Intuitive Approach) PURPOSE: To introduce the notion of a limit. This section introduces the idea of a limit by discussing how to find the slope of a tangent line to a curve. This is shown using the slopes of secant lines to a function (lines through two points on the function) as the points become closer together....
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This note was uploaded on 06/01/2010 for the course CAL 1000 taught by Professor Lee during the Spring '10 term at École Normale Supérieure.
 Spring '10
 lee

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