Limits and Their Properties

Transcription

1 Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus is mathematics the mathematics of change, that is, the of velocities, accelerations, tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations. Instructor Date How to understand what calculus is and how it compares with precalculus List some problem-solving strategies that will be helpful in the study of calculus. Be sure to understand the question. Outline a plan. (Some approaches that might be useful include: look for a pattern, solve a simpler problem, work backwards, draw a diagram, or use technology.) Complete your plan. (Be sure to answer the question.) Look back at your work. Does your answer make sense? II. The Tangent Line Problem (Page 45) In the tangent line problem, you are given a function f and a point P on its graph and are asked to find an equation of the tangent line to the graph at point P. How to understand that the tangent line problem is basic to calculus Ecept for cases involving a vertical tangent line, the problem of finding the tangent line at a point P is equivalent to finding the slope of the tangent line at P. You can approimate this slope by using a line through the point of tangency and a second point on the curve. Such a line is called a secant line. 17

2 18 Chapter 1 Limits and Their Properties If P(c, f(c)) is the point of tangency and Q(c + Δ, f(c + Δ)) is a second point on the graph of f, the slope of the secant line through these two points can be found using precalculus and is f(c + Δ) f(c) given by m sec. Δ As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line. When such a limiting position eists, the slope of the tangent line is said to be the limit of the slope of the secant line. III. The Area Problem (Page 46) A second classic problem in calculus is finding the area of a plane region that is bounded by the graphs of functions. This problem can also be solved with a limit process. In this case, the limit process is applied to the area of a rectangle to find the area of a general region. How to understand that the area problem is also basic to calculus Consider the region bounded by the graph of the function y f( ), the -ais, and the vertical lines a and b. You can approimate the area of the region with several rectangular regions. As you increase the number of rectangles, the approimation tends to become better and better because the amount of area missed by the rectangles decreases. Your goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increases without bound. Homework Assignment Page(s) Eercises

3 Section 1.2 Finding Limits Graphically and Numerically 19 Section 1.2 Finding Limits Graphically and Numerically Objective: In this lesson you learned how to find limits graphically and numerically. Course Number Instructor Date I. An Introduction to Limits (Pages 48 49) The notation for a limit is lim f ( ) L, which is read as the limit of f() as approaches c is L. How to estimate a limit using a numerical or graphical approach The informal description of a limit is as follows: If f() becomes arbitrarily close to a single number L as approaches c from either side, the limit of f(), as approaches c, is L Describe how to estimate the limit lim numerically Let f () ( )/( + 2). Then construct a table that shows values of f () for two sets of -values one set that approaches 2 from the left and one that approaches 2 from the right. Use the table to look for a numerical trend in the value of f () as approaches 2. This is an estimate of the limit. 2 The eistence or noneistence of f() at c has no bearing on the eistence of the limit of f() as approaches c. II. Limits That Fail to Eist (Pages 50 51) If a function f() approaches a different number from the right side of c than it approaches from the left side, then the limit of f() as approaches c does not eist. How to learn different ways that a limit can fail to eist If f() is not approaching a real number L that is, if f() increases or decreases without bound as approaches c, you can conclude that the limit does not eist. The limit of f() as approaches c also does not eist if f() oscillates between two fied values as approaches c.

4 20 Chapter 1 Limits and Their Properties III. A Formal Definition of Limit (Pages 52 54) The ε-δ definition of limit assigns mathematically rigorous meanings to the two phrases f() becomes arbitrarily close to L and approaches c used in the informal description of limit. How to study and use a formal definition of limit Let ε represent a small positive number. Then the phrase f() becomes arbitrarily close to L means that f() lies in the interval (L ε, L + ε). Using absolute value, you can write this as f() L < ε. The phrase approaches c means that there eists a positive number δ such that lies in either the interval (c δ, c) or the interval (c, c + δ). This fact can be concisely epressed by the double inequality 0 < c < δ. State the formal ε-δ definition of limit. Let f be a function defined on an open interval containing c (ecept possibly at c) and let L be a real number. The statement lim f() L means that for each ε > 0 there eists a δ > 0 such that if 0 < c < δ, then f() L < ε. Eample 1: Use the ε-δ definition of limit to prove that lim (10 3 ) Homework Assignment Page(s) Eercises

5 Section 1.3 Evaluating Limits Analytically 21 Section 1.3 Evaluating Limits Analytically Objective: In this lesson you learned how to evaluate limits analytically. Course Number Instructor Date I. Properties of Limits (Pages 59 61) The limit of f() as approaches c does not depend on the value of f at c. However, it may happen that the limit is precisely f(c). In such cases, the limit can be evaluated by direct substitution. How to evaluate a limit using properties of limits Theorem 1.1 Let b and c be real numbers and let n be a positive integer. Complete each of the following properties of limits. 1. b 2. lim c 3. lim c n n Theorem 1.2 Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. lim f ( ) L and lim g( ) K Complete each of the following statements about operations with limits. 1. Scalar multiple: lim[ b f ( )] bl 2. Sum or difference: lim[ f ( ) ± g( )] 3. Product: lim[ f ( ) g( )] L ± K LK f ( ) 4. Quotient: lim c g ( ) L/K, provided K 0 5. Power: lim [ f ( )] L n n Eample 1: Find the limit: 48 lim

6 22 Chapter 1 Limits and Their Properties The limit of a polynomial function p() as is simply the value of p at c. This direction substitution property is value for all polynomial and rational functions with nonzero denominators. Theorem 1.3 If p is a polynomial function and c is a real number, then lim p( ) p(c). If r is a rational function given by r ( ) p ( )/ q ( ) and c is a real number such that qc ( ) 0, then lim r ( ) r(c) p(c)/q(c). Theorem 1.4 Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even: lim n n c Theorem 1.5 If f and g are functions such that lim g ( ) L and lim f ( ) f ( L ), then lim f ( g ( )) f (lim g()) f(l). L Theorem 1.6 Let c be a real number in the domain of the given trigonometric function. Complete each of the following limit statements. 1. lim sin sin c 2. lim cos cos c 3. lim tan tan c 4. lim cot cot c 5. lim sec sec c 6. lim csc csc c Eample 2: Find the following limits. 4 2 a. lim b. limcos π a. 3 b. 1

7 Section 1.3 Evaluating Limits Analytically 23 II. A Strategy for Finding Limits (Page 62) Theorem 1.7 Let c be a real number and let f ( ) g( ) for all c in an open interval containing c. If the limit of g() as approaches c eists, then the limit of f() also eists and lim f ( ) lim g(). How to develop and use a strategy for finding limits This theorem states that if two functions agree at all but a single point c, then they have identical limit behavior at c. List four steps in the strategy for finding limits. 1. Learn to recognize which limits can be evaluated by direct substitution. 2. If the limit of f() as approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all other than c. [Choose g such that the limit of g() can be evaluated by direct substitution.] 3. Apply Theorem 1.7 to conclude analytically that lim f() lim g() g(c). 4. Use a graph or table to reinforce your conclusion. III. Dividing Out and Rationalizing Techniques (Pages 63 64) An epression such as the meaningless fractional form 0/0 is called a(n) indeterminate form because you cannot, from the form alone, determine the limit. When you try to evaluate a limit and encounter this form, remember that you must rewrite the fraction so that the new denominator does not have 0 as its limit. One way to do this is to divide out like factors, using the dividing out technique. Another technique is to rationalize the numerator. How to evaluate a limit using dividing out and rationalizing techniques Eample 3: Find the following limit: 2 lim

8 24 Chapter 1 Limits and Their Properties If you apply direct substitution to a rational function and obtain pc () 0 rc (), then by the Factor Theorem of Algebra, you qc () 0 can conclude that ( c) must be a common factor to both p() and q(). IV. The Squeeze Theorem (Pages 65 66) Theorem 1.8 The Squeeze Theorem If h ( ) f( ) g ( ) for all in an open interval containing c, ecept possibly at c itself, and if lim h ( ) L lim g ( ), then lim f ( ) eists and is equal to L. How to evaluate a limit using the Squeeze Theorem Theorem 1.9 Two Special Trigonometric Limits sin lim cos lim 0 0 y y y Homework Assignment Page(s) Eercises

9 Section 1.4 Continuity and One-Sided Limits 25 Section 1.4 Continuity and One-Sided Limits Objective: In this lesson you learned how to determine continuity at a point and on an open interval, and how to determine one-sided limits. Course Number Instructor Date Important Vocabulary Define each term or concept. Discontinuity If a function f is defined on an open interval I and f is not continuous at c, then f is said to have a discontinuity at c. Greatest integer function f ( ) less than or equal to. The function defined as the greatest integer I. Continuity at a Point and on an Open Interval (Pages 70 71) To say that a function f is continuous at c means that there is no interruption in the graph of f at c: the graph is How to determine continuity at a point and continuity on a open interval unbroken and there are no holes, jumps, or gaps. A function f is continuous at c if the following three conditions are met: 1. f(c) is defined 2. lim f() eists 3. lim f() f(c) If f is continuous at each point in the interval (a, b), then it is continuous on an open interval (a, b). A function that is continuous on the entire real line (, ) is everywhere continuous. A discontinuity at c is called removable if f can be made continuous by appropriately defining or. redefining f(c).

10 26 Chapter 1 Limits and Their Properties A discontinuity at c is called nonremovable if f cannot be made continuous at c by defining or redefining the. function at c. II. One-Sided Limits and Continuity on a Closed Interval (Pages 72 74) A one-sided limit is the limit of a function f() at c from either just the left of c or just the right of c. How to determine onesided limits and continuity on a closed interval lim f ( ) L is a one-sided limit from the right and means + that approaches c from values greater than c.. lim f ( ) L is a one-sided limit from the left and means that approaches c from values less than c.. One-sided limits are useful in taking limits of functions involving radicals. When the limit from the left is not equal to the limit from the right, the (two-sided) limit does not eist. Let f be defined on a closed interval [a, b]. If f is continuous on the open interval (a, b) and lim f ( ) f( a) and lim f ( ) f( b), + a b then f is continuous on the closed interval [a, b]. Moreover, f is continuous from the right at a and continuous from the left at b. III. Properties of Continuity (Pages 75 76) If b is a real number and f and g are continuous at c, then the following functions are also continuous at c. How to use properties of continuity 1. Scalar multiple: bf 2. Sum or difference: f ± g 3. Product: fg 4. Quotient: f/g, if g(c) 0

11 Section 1.4 Continuity and One-Sided Limits 27 A polynomial function is continuous at number. A rational function is continuous at domain. every real every number in its If g is continuous at c and f is continuous at g(c), then the composite function given by ( f o g)( ) f( g( )) is continuous at c. IV. The Intermediate Value Theorem (Pages 77 78) Intermediate Value Theorem If f is continuous on the closed interval [a, b], f ( a) f( b), and k is any number between f(a) and How to understand and use the Intermediate Value Theorem f(b), then there is at least one number c in [a, b] such that f(c) k. Eplain why the Intermediate Value Theorem is called an eistence theorem. The Intermediate Value Theorem tells you that at least one c eists, but it does not give a method for finding c. The Intermediate Value Theorem states that for a continuous function f, if takes on all values between a and b, f() must take on all values between f(a) and f(b). The Intermediate Value Theorem often can be used to locate zeros of a function that is continuous on a closed interval. Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the eistence of at least one zero of f in the closed interval [a, b].

12 28 Chapter 1 Limits and Their Properties Eplain how the bisection method can be used to approimate the real zeros of a continuous function. If you know that a zero eists in the closed interval [a, b], the zero must lie in the interval [a, (a + b)/2] or [(a + b)/2, b]. From the sign of f([a + b]/2), you can determine which interval contains the zero. By repeatedly bisecting the interval, you can close in on the zero of the function. Additional notes y y y Homework Assignment Page(s) Eercises

13 Section 1.5 Infinite Limits 29 Section 1.5 Infinite Limits Objective: In this lesson you learned how to determine infinite limits and find vertical asymptotes. Course Number Instructor Date I. Infinite Limits (Pages 83 84) A limit in which f() increases or decreases without bound as approaches c is called an infinite limit. How to determine infinite limits from the left and from the right Let f be a function that is defined at every real number in some open interval containing c (ecept possibly at c itself). The statement lim ( ) means that for each M > 0 there eists a δ > 0 such that f() > M whenever 0 < c < δ. Similarly, the statement lim ( ) means that for each N < 0 there eists a δ > 0 such that f() < N whenever 0 < c < δ. To define the infinite limit from the left, replace 0 < c < δ by c δ < < c. To define the infinite limit from the right, replace 0 < c < δ by c < < c + δ. Be sure to see that the equal sign in the statement lim f( ) does not meant that the limit eists! On the contrary, it tells you how the limit fails to eist by denoting the unbounded behavior of f() as approaches c. II. Vertical Asymptotes (Pages 84 87) If f() approaches infinity (or negative infinity) as approaches c from the right or the left, then the line c is a vertical asymptote of the graph of f. How to find and sketch the vertical asymptotes of the graph of a function Let f and g be continuous on an open interval containing c. If f() c 0, gc ( ) 0, and there eists an open interval containing c such that g ( ) 0 for all c in the interval, then the graph

14 30 Chapter 1 Limits and Their Properties of the function given by f ( ) h ( ) has a vertical g( ) asymptote at c. If both the numerator and denominator are 0 at c, you obtain the indeterminate form 0/0 and you cannot determine the limit behavior at c without further investigation, such as simplifying the epression. Eample 1: Determine all vertical asymptotes of the graph of f( ) Theorem 1.15 Let c and L be real numbers and let f and g be functions such that lim f ( ) and lim g ( ) L Complete each of the following statements about operations with limits. 1. Sum or difference: lim[ f ( ) ± g( )] 2. Product: lim[ f ( ) g( )] lim[ f ( ) g( )], L > 0, L < 0 3. Quotient: g( ) lim 0 f ( ) Eample 2: Determine the limit: 1 lim Homework Assignment Page(s) Eercises