Mathematics
Limits of a Function
In mathematics, the limit of a function describes the behavior of the function as the input approaches a certain value. It represents the value that the function approaches as the input gets closer and closer to the specified value. Limits are fundamental in calculus and are used to define continuity, derivatives, and integrals.
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12 Key excerpts on "Limits of a Function"
eBook - PDF- Robert G. Bartle, Donald R. Sherbert(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
CHAPTER 4 LIMITS ‘‘Mathematical analysis’’ is generally understood to refer to that area of mathematics in which systematic use is made of various limiting concepts. In the preceding chapter we studied one of these basic limiting concepts: the limit of a sequence of real numbers. In this chapter we will encounter the notion of the limit of a function. The rudimentary notion of a limiting process emerged in the 1680s as Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) struggled with the creation of the Calculus. Though each person’s work was initially unknown to the other and their creative insights were quite different, both realized the need to formulate a notion of function and the idea of quantities being ‘‘close to’’ one another. Newton used the word ‘‘fluent’’ to denote a relationship between variables, and in his major work Principia in 1687 he discussed limits ‘‘to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.’’ Leibniz introduced the term ‘‘function’’ to indicate a quantity that depended on a variable, and he invented ‘‘infinitesimally small’’ numbers as a way of handling the concept of a limit. The term ‘‘function’’ soon became standard terminology, and Leibniz also introduced the term ‘‘calculus’’ for this new method of calculation. In 1748, Leonhard Euler (1707–1783) published his two-volume treatise Introduc- tio in Analysin Infinitorum, in which he discussed power series, the exponential and logarithmic functions, the trigonometric functions, and many related topics. This was followed by Institutiones Calculi Differentialis in 1755 and the three-volume Institu- tiones Calculi Integralis in 1768–1770. These works remained the standard textbooks on calculus for many years. But the concept of limit was very intuitive and its looseness led to a number of problems.
eBook - PDF- Steven G. Krantz(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 5 Limits and Continuity of Functions 5.1 Definition and Basic Properties of the Limit of a Function Preliminary Remarks Questions about limits go back to the ancient Greeks. The Greeks really did not understand limits (witness Zeno’s paradoxes). The question of limits arose even more intensely in the development of calculus. Isaac Newton did not understand limits, and neither did Leibniz. It took the combined efforts of a number of nineteenth-century mathematical geniuses—including Cauchy, Riemann, Dirichlet, Weierstrass, and others—to finally nail down the concept of limit. Here we present the fruits of their efforts. In this chapter we are going to treat some topics that you have seen before in your calculus class. However, we shall use the deep properties of the real numbers that we have developed in this text to obtain important new insights. Therefore you should not think of this chapter as review. Look at the concepts introduced here with the power of your new understanding of analysis. Definition 5.1 Let f be a real-valued function whose domain E contains adjoining intervals ( a,c ) and ( c,b ). Let lscript be a real number. We say that lim x → c f ( x ) = lscript 101 102 CHAPTER 5. LIMITS AND CONTINUITY OF FUNCTIONS y x Figure 5.1: The limit of an oscillatory function. if, for each epsilon1> 0, there is a δ> 0 such that, when 0 < | x -c | <δ , then | f ( x ) -lscript |
eBook - PDF- P Adams, K Smith;R V??born??;;(Authors)
- 2004(Publication Date)
- WSPC(Publisher)
Chapter 12 Limits and Continuity of Functions Limits of functions are defined in terms of limits of sequences. With a function f continuous on an interval we associate the in- tuitive idea of the graph f being drawn without lifting the pencil from the drawing paper. Mathematical treatment of continuity starts with the definition of a function continuous at a point; this definition is given here in terms of a limit of a function at a point. In this chapter we shall develop the theory of limits of functions, study continuous functions, and particularly functions continuous on closed bounded intervals. At the end of the chapter we touch upon the concept of limit superior and inferior of a function. 12.1 Limits X Looking at the graph of f : x w -(l - x) (Figure 12.1), it is natural to say that the function value approaches 1 as x approaches 0 from the right. 1x1 Formally we define: efinition 12.1 A function f, with domf c R, is said to have a limit 1 at 2 from the right if for every sequence n xn for which 2, + 2 and x, > 5 it follows that f(xn) + 1 . If f has a limit 1 at 2 from the right, we write limf(x) = 1. X l a : Definition 12.2 If the condition xn > 2 is replaced by xn < 2 one obtains the definition of the limit of f at 2 from the left. The limit of ( f at 2 from the left is denoted by limf(x). xt2 J Remark 12.1 The symbols x J, 2 and x 2 can be read as “x decreases 313 314 Introduction to Mathematics with Maple to k” and “x increases to 2” , respectively. -1.5 -1 -0.5 -0.5 - X Fig. 12.1 Graph of f(z) = -(1 - x) 1x1 Example 12.1 the beginning of this section; that is, We now prove, according to Definition 12.1, the limit from X lim -(I - x) = 1. X l O 1x1 lxn I X n If X n + 0 and xn > 0 then -(1 - xn) = (1 - xn) + 1. Similarly, X X n lim-(1 - x) = -1 because -(1 - xn) = xn - 1 for xn < 0 and since xto 1x1 lxnl xn - 1 + -1 if xn 4 0. Remark 12.2 It was natural to say in Example 12.1 that f has a limit from the right a t 0 even though f was not defined a t 0.
eBook - PDFMathematics
An Applied Approach
- Michael Sullivan, Abe Mizrahi(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
We may describe the meaning of as follows: For all values of x approximately equal to c, with x c, the corresponding value f (x) is approximately equal to N. lim x c f (x) N lim x c f (x) N Finding Limits Using Tables and Graphs 677 A L O O K B AC K , A L O O K F O RWA R D In Chapter 10 we defined a function and many of the proper- ties that functions can have. In Chapter 11 we discussed class- es of functions and properties that the classes have. With this as background we are ready to study the limit of a function. This concept is the bridge that takes us from the mathematics of algebra and geometry to the mathematics of calculus. Calculus actually consists of two parts: the differential cal- culus, which we discuss in Chapters 13 and 14 and the integral calculus, discussed in Chapters 15 and 16. In Chapter 17 we study the calculus of functions of two or more variables. In differential calculus we introduce another property of functions, namely the derivative of a function. We shall find that the derivative opens up a way for doing many applied problems in business, economics, and social sciences. Many of these applications involve an analysis of the graph of a function. PREPARING FOR THIS SECTION Before getting started, review the following: 12.1 Finding Limits Using Tables and Graphs > Evaluating Functions (Chapter 10, Section 10.2, pp. 550 – 551) > Piecewise-defined Functions (Chapter 10, Section 10.4, pp. 583 – 585) > Library of Functions (Chapter 10, Section 10.4, pp. 579 – 583) OBJECTIVES 1 Find a limit using a table 2 Find a limit using a graph 678 Chapter 12 The Limit of a Function From Table 1, we infer that as x gets closer to 3 the value of f (x) 5x 2 gets closer to 45. That is, ◗ When choosing the values of x in a table, the number to start with and the subse- quent entries are arbitrary. However, the entries should be chosen so that the table makes it clear what the corresponding values of f are getting close to.
eBook - PDFCounterExamples
From Elementary Calculus to the Beginnings of Analysis
- Andrei Bourchtein, Ludmila Bourchtein(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 7 Limits and continuity General Remark . Part II (chapters 7-9) contains the topics of the func-tions of two variables, which are quite representative for the most topics of the multivariable functions. The counterexamples presented in this part of the work are separated into two groups. The first one includes those examples that have intimate connections with counterexamples of one-variable functions considered in Part I. They show how the ideas applied in the one-dimensional case can be generalized/extended to many variables. Accordingly, the exam-ples of this group are placed in the sections titled “one-dimensional links” in each of chapters 7-9. The examples of the second group have a weak or no connection with one-dimensional case, highlighting a specificity of concepts and results for multivariable functions. Some of them illustrate the situations that are feasible for two-variable functions but cannot happen in the case of one-variable functions. In each chapter, all the examples of the second group are collected in sections titled “multidimensional essentials”. 7.1 Elements of theory Limits. Concepts Limit (general limit) . Let f ( x, y ) be defined on X and ( a, b ) be a limit point of X . We say that the limit of f ( x, y ), as ( x, y ) approaches ( a, b ), exists and equals A if for every ε > 0 there exists δ > 0 such that for all ( x, y ) ∈ X such that 0 < √ ( x − a ) 2 + ( y − b ) 2 < δ it follows that | f ( x, y ) − A | < ε . The usual notations are lim ( x,y ) → ( a,b ) f ( x, y ) = A and f ( x, y ) → ( x,y ) → ( a,b ) A . Remark . In calculus, a non-essential simplification that f ( x, y ) is defined in some deleted neighborhood of ( a, b ) is frequently used. 207 208 Counterexamples: From Calculus to the Beginnings of Analysis Partial limit . Let f ( x, y ) be defined on X , S be a subset of X and ( a, b ) be a limit point of S .
eBook - PDFBrief Calculus
An Applied Approach
- Michael Sullivan(Author)
- 2021(Publication Date)
- Wiley(Publisher)
In Chapter 8 we study the calculus of nctions of two or more variables. In differential calculus we introduce another property of nctions, namely the derivative of a function. We shall d that the derivative opens up a way r doing many applied problems in business, economics, and social sciences. Many of these applications involve an analysis of the graph of a nction. m Finding Limits Using Tables and Graphs PREPARING FOR THIS SECTION Before getting started, review the following: > Evaluating Functions (Chapter 1. Section 1.2. pp. 108-112) > Piecewise-defined Functions (Chapter 1. Section 1.4. pp. 143-145) OBJECTIVES 1 Find a limit using a table 2 Find a limit using a graph > Library of Functions (Chapter 1. Section 1.4. pp. 137-143) The idea of the limit of a nction is what connects algebra and geometry to calculus. In working with the limit of a nction, we encounter notation of the rm limf(x) = N xc This is read as "the limit of f(x) as x approaches c equals the number N" Here f is a nction defined on some open interval containing the number c; f need not be defined at c, however. may describe the meaning of lim f(x) = N as llows: xc For all values of x approximately equal to c, with x c, the corresponding value f(x) is approximately equal to N 238 Chapter 3 The Limit of a Function TABLE 2 X Y1 2.9 .71 2.99 .97 2199 .997 3.11 .3 31 .3 .301 X= EXAMPLE 1 Another description of lim f(x) = N is xc As x gets closer to c, but remains unequal to c, the corresponding value of f(x) gets closer to N. bles generated with the help of a calculator are usel r finding limits. Finding a Limit Using a Table Find: lim(5x 2 ) x3 SOLUTION Here f(x) = 5x2 and c = 3. We choose values of x close to 3, arbitrarily starting with 2.99. Then we select additional numbers that get closer to 3, but remain less than 3. Next we choose values of x greater than 3, starting with 3.01, that get closer to 3.
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 3 THEORY OF LIMITS § 1. The limit of a function 26. Historical remarks. The concept of limit now enters into the whole of mathematical analysis and also plays an important part in other branches of mathematics. However (as the reader will see in Chapter 14), this concept was certainly not the basis of the differential and integral calculus at the time of their creation. The concept of a limit appears for the first time (essentially in the same form as it will be given below in Sec. 28) in the works of Wallist in his Arithmetic of Infinite Quantities (1655). Newton in the celebrated Mathematical Foundations of Natural Philosophy (1686-1687) announced his method of the first and last ratios (sums) in which the beginnings of the theory of limits can be seen. However, none of the great mathematicians of the eighteenth century tried to base the new calculus on the concept of limit and by doing so to meet the just criticism to which the calculus was subject*. In this respect Euler's views are characteristic; in the foreword to his treatise on Differential Calculus (1755) he clearly speaks of the limit but nowhere in the book makes use of this concept. The turning point in this problem is due to the Algebraic Analysis (1821) of Cauchy§ and his further publications, in which for the first time the theory of limits was developed; it was used by Cauchy as an effective means to a precise construc-tion of mathematical analysis. Cauchy's standpoint, which destroyed the mystique surrounding the foundations of analysis, was widely recognized. Strictly speaking, Cauchy's merit is shared also by other scholars—particularly Bolzano; in many cases his papers were prior to those of Cauchy and later mathe-maticians. They however were not known at the time and were remembered only after many decades.
eBook - PDF- Laura R. Van Zoest(Author)
- 2003(Publication Date)
- Information Age Publishing(Publisher)
For the dynamic-practical statement (see Table 3.1) she said, “that’s what I did each time.” Alice believed that the limit was a specific value, but that it could be reached at several x -values. “As x is approaching a certain point, the function is either 1 x + ( ) 1 x --x 0 → lim 52 W. J. HARRINGTON remaining stagnant or it is changing. No matter what I do, once I get to a certain point, as soon as my x reaches a certain point, it [the function value] just stays at that point.” She described limit as how the function moved. When the movement ceased, meaning the function became con-stant (in her eyes), she was convinced that the limit had been found. So while Alice believed a limit to be a static “target” value as defined in her Applied Calculus class, her conception is better described as an inaccu-rate dynamic conception. It also seems clear that her particular dynamic conception is a product of her solution procedure. In the following exchange Carrie provides another example of the dynamic conception as she explains the concept of limit. Bill: How would you explain to someone who doesn’t know anything about limits, what a limit is? Carrie: A limit is um, if you look at a function, it’s the value that all the other values are approaching. Like if you keep on plugging in x -values, you’ll keep on getting y -values that are gonna approach…Okay as all these x-values are approaching the x , then all the y -values are gonna approach f ( x ). All the f(x ) values are gonna approach f ( x ). Does that make sense? Bill: What do you mean by f ( x )? Carrie: f ( x ) is a function of x . Bill: How does that relate to the limit? Carrie: Okay, the limit is um, a certain value, like a certain, I guess value, as um, all other, like when you use x you plug it into a function and you’ll get an answer which is f ( x ).
- Mark Ryan(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
As you drag the line to the right, that point travels along the function, winding up and down along the road, and, as you drag the line over the origin, the point reaches and then passes 0, 0 . Now tell me this: When the point hits 0, 0 , is it on its way up or down? How can you reconcile all this? I wish I knew. Stuff like this really messes with your mind. 158 UNIT 3 Limits You can see on the graph (in the first quadrant) that as x gets bigger and bigger — in other words, as x approaches infinity — the height of the function gets lower and lower but never gets to zero. This is confirmed by considering what happens when you plug bigger and bigger numbers into 1 x : The outputs get smaller and smaller and approach zero. This graph thus has a horizontal asymptote of y 0 (the x-axis), and you say that lim x x 1 0. The fact that x never actually reaches infinity and that f never gets to zero has no relevance. When we say that lim x x 1 0, we mean that as x gets bigger and bigger without end, f is closing in on a height of zero (or f is ultimately getting infinitely close to a height of zero). If you look at the third quadrant, you can see that the function f also approaches zero as x approaches negative infinity, which is written as lim x x 1 0. Like with limits where x approaches a finite number, to solve limit problems where x approaches infinity or negative infinity, you can use your calculator or algebra. But before I go through those techniques, let’s first take care of a special class of limits at infinity where no calculus is needed: rational function limits. Limits of rational functions at ± infinity This section deals with the horizontal asymptotes of rational functions. A rational function is a fraction function with polynomials in the numerator and denominator.
eBook - PDFPrecalculus
Functions and Graphs
- Earl Swokowski, Jeffery Cole(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The preceding limit can also be established by means of the definition of a limit of a function. We state these facts for reference in the next theorem. Techniques for Finding Limits 11.3 Theorem on Limits (1) lim x l a c 5 c (2) lim x l a x 5 a 808 CHAPTER 11 Limits of Functions Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. As we shall see, the limits in the preceding theorem can be used as build-ing blocks for finding limits of very complicated expressions. I L L U S T R A T I O N ■ lim x l 3 8 5 8 ■ lim x l 8 3 5 3 ■ lim x l Ï 2 x 5 Ï 2 ■ lim x l 2 4 x 5 2 4 Many functions can be expressed as sums, differences, products, and quotients of other functions. Suppose f and g are functions and L and M are real numbers. If f s x d l L and g s x d l M as x l a , we would expect that f s x d 1 g s x d l L 1 M as x l a . The next theorem states that this expectation is true and gives analogous results for products and quotients. We may state properties (3)–(7) above as follows: (3) The limit of a sum is the sum of the limits. (4) The limit of a product is the product of the limits. (5) The limit of a quotient is the quotient of the limits, provided the denomi-nator has a nonzero limit. (6) The limit of a constant times a function is the constant times the limit of the function. (7) The limit of a difference is the difference of the limits. Proofs for (3)–(5), based on the definition of a limit of a function, are given in Appendix V.
eBook - PDFCalculus
Resequenced for Students in STEM
- David Dwyer, Mark Gruenwald(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
61. Let f (x) = x 3 - 9x + 1. Show that f has a zero on the interval [1, 3] 62. Show that the equation x 2 +1 = 3 sin x has a solution in the interval [0, π]. 63. Show that there is a negative number x for which x 2 = e x . 64. Let g(x) = cos x + sin x. Show that g(x) = 1 2 for some value of x on the interval [0, π]. 2.6. FORMAL DEFINITION OF LIMIT 101 Applications Exercises 65–68 Determine whether or not the given function is continuous over its domain. 65. Vehicle Distance f (t) is the distance traveled in t hours by a car whose velocity varies between 0 and 60 miles per hour. 66. Postage Cost g(x) is the amount of postage required by the United States Postal Service for a first-class letter weighing x ounces. 67. Parking Cost f (x) is the cost to park a vehicle in a lot for x minutes if the lot charges $5 for the first hour and $2 for each additional 30 minutes, up to a daily maximum of $20. 68. Temperature T (t) is the temperature at a specific loca- tion t hours after midnight on a specific day. 2.6 Formal Definition of Limit Most of our key results involving limits have been deduced by appealing to intu- ition or common sense. But in mathematics, common sense can fail us, particularly when it comes to the topic of limits. For example, intuition may suggest to you that 0. ¯ 9 (that is, 0.9999 . . .) is less than 1; but in fact, we can prove that 0. ¯ 9 is equal to 1. Because limits are the foundation on which calculus is built, it’s particularly important to see that there are no gaps or logical leaps in its development, no room for disagreement. Our approach in this section is to begin with a careful definition of limits of sequences. Then we will use limits of sequences as a tool to define limits of general functions. This will lead us to an equivalent definition of the limit of a function which doesn’t involve sequences. Limits of Sequences When we say that lim n→∞ a n = L, we mean that a n can be made as close to L as we please by making n large enough.
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
1. The function is not defined at x = c. 2. The limit of f (x) does not exist at x = c. 3. The limit of f (x) exists at x = c, but it is not equal to f (c). If none of the three conditions is true, then the function f is called continuous at c, as indicated in the important definition below. Definition of Continuity Continuity at a Point A function f is continuous at c when these three conditions are met. 1. f (c) is defined. 2. lim xuni2192c f (x) exists. 3. lim xuni2192c f (x) = f (c) Continuity on an Open Interval A function is continuous on an open interval parenleft.alt1a, bparenright.alt1 when the function is continuous at each point in the interval. A function that is continuous on the entire real number line (- ∞ , ∞ ) is everywhere continuous. FOR FURTHER INFORMATION For more information on the concept of continuity, see the article “Leibniz and the Spell of the Continuous” by Hardy Grant in The College Mathematics Journal. To view this article, go to MathArticles.com. Exploration Informally, you might say that a function is continuous on an open interval when its graph can be drawn with a pencil without lifting the pencil from the paper. Use a graphing utility to graph each function on the given interval. From the graphs, which functions would you say are continuous on the interval? Do you think you can trust the results you obtained graphically? Explain your reasoning. Function Interval a. y = x 2 + 1 (-3, 3) b. y = 1 x - 2 (-3, 3) c. y = sin x x (-π , π ) d. y = x 2 - 4 x + 2 (-3, 3) 2.4 Continuity and One-Sided Limits 95 Consider an open interval I that contains a real number c. If a function f is defined on I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. Discontinuities fall into two categories: removable and nonremovable. A discontinuity at c is called removable when f can be made continuous by appropriately defining (or redefining) f (c).
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