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Stanford MATH 19 - Lecture Outline

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Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (More Limit Laws and Continuity)Friday, January 18AnnouncementsMonday’s office hours will be canceled in honor of MLK day.Quiz 1 will be at 9:00 on Wednesday, January 23.Homework 1 will be due at 9:15 in class on Wednesday.The course assistant, Ziyu, has office hours on Tuesday evening from 7 to 10.RecapRecall from Wednesday’s lecture the limit laws:1. limx→ac = c when c and a are a constants.2. limx→ax = a when a is a constant.3. When limx→af(x) and limx→ag(x) exist, so does the limit of their sum, difference, product,and division (provided limx→ag(x) 6= 0.)4. The substitution rule for polynomials and rational functions. That is, if f(x) is a polynomialor a rational function and f (x) is defined at a then limx→af(x) = f(a).Also recall the definitions of the left- and right-hand limits.One more limit lawlimx→anpf(x) =nqlimx→af(x)where n is a positive integer. If n is even, we require that limx→af(x) > 0.Theory: Left- and Right-hand Limits CtdImportant Fact: limx→af(x) = L if and only if limx→a−f(x) = L and limx→a+f(x) = L.Example: Letf(x) = |x| =x : x ≥ 0−x : x < 0Where does the limit exist and what is it?1ContinuityAs we said b e fore, limx→af(x) is not affected by the value of f(a). But it is tempting to just eval-uate f(a) to find this limit. This does not always work, but we can make the following statements.Definition. If a function statisfies limx→af(x) = f(a), we say that f(x) is continuous atx = a. If f(x) is continuous at every point in its domain, we s ay that f(x) is a continuousfunction.Polynomials, rational functions, ro ot functions, trigonometric functions, inverse trigonometricfunctions, exponential functions, and logarithmic functions are continuous functions on their do-mains.Useful fact: If f(x) is a combination (i.e. sum, difference, product, division, or composi-tion) of continuous functions, then f(x) is continuous on its domain. There fore, in these cases,limx→af(x) = f(a).Examples of continuous functions1. What is limx→4f(x)?2. What is limx→1e√ln(x)− cos(sin(πx))?3. What is limx→1x11+1/x√1−x+x2?What to Know/Memorize1. limx→af(x) = L if and only if limx→a−f(x) = L and limx→a+f(x) = L.2. The definition of a continuous function and how to find the limit of a continuous function ata point in its


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Stanford MATH 19 - Lecture Outline

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