Limit Laws

10 Key excerpts on "Limit Laws"

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  eBook - PDF

  Biocalculus

  Calculus, Probability, and Statistics for the Life Sciences

  • James Stewart, Troy Day, James Stewart(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  ■ The Limit Laws The following five properties of limits are similar to the ones we have seen before, but some of the other properties of limits apply only when the variable x approaches a finite number. Suppose that c is a constant and the limits lim x l a f s x d and lim x l a t s x d exist. Then 1. lim x l a f f s x d 1 t s x dg -lim x l a f s x d 1 lim x l a t s x d 2. lim x l a f f s x d 2 t s x dg -lim x l a f s x d 2 lim x l a t s x d 3. lim x l a f cf s x dg -c lim x l a f s x d 4. lim x l a f f s x d t s x dg -lim x l a f s x d ? lim x l a t s x d 5. lim x l a f s x d t s x d -lim x l a f s x d lim x l a t s x d if lim x l a t s x d ± 0 These laws can be stated verbally as follows: 1. The limit of a sum is the sum of the limits. 2. The limit of a difference is the difference of the limits. 3. The limit of a constant times a function is the constant times the limit of the function. 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 that the limit of the denominator is not 0). It is easy to believe that these properties are true. For instance, if f s x d is close to L and t s x d is close to M , it is reasonable to conclude that f s x d 1 t s x d is close to L 1 M . This gives us an intuitive basis for believing that Law 1 is true. All of these laws can be proved using the precise definition of a limit. In Appendix E we give the proof of Law 1. Limit Laws for Functions Sum Law Difference Law Constant Multiple Law Product Law Quotient Law Copyright 2016 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.

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  Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. In Section 2.2 we used calculators and graphs to guess the values of limits, but we saw that such methods don’t always lead to the correct answer. In this section we use the following properties of limits, called the Limit Laws, to calculate limits. Limit Laws Suppose that is a constant and the limits exist. Then 1. 2. 3. 4. 5. These five laws can be stated verbally as follows: Sum Law 1. The limit of a sum is the sum of the limits. Difference Law 2. The limit of a difference is the difference of the limits. Constant Multiple Law 3. The limit of a constant times a function is the constant times the limit of the function. Product Law 4. The limit of a product is the product of the limits. Quotient Law 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0). It is easy to believe that these properties are true. For instance, if is close to and is close to , it is reasonable to conclude that is close to . This gives us an intuitive basis for believing that Law 1 is true. All of these laws can be proved using the precise definition of a limit. In Appendix E we give the proof of Law 1. L M f x t x M t x L f x lim x l a f x t x lim x l a f x lim x l a t x if lim x l a t x 0 lim x l a f x t x lim x l a f x lim x l a t x lim x l a cf x c lim x l a f x lim x l a f x t x lim x l a f x lim x l a t x lim x l a f x t x lim x l a f x lim x l a t x lim x l a t x and lim x l a f x c 104 CHAPTER 2 LIMITS AND DERIVATIVES crosses the y -axis to estimate the limit of as x approaches 0.

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  7

  LIMITS

  7.1INTRODUCTION

   

  The concept of a limiting value of a function plays an important role in calculus, because the formal definition of the derivative of a function at a point in its domain can be expressed as the limiting value of a particular expression involving the function.

  The meaning of “limit” in mathematics is more subtle than that in everyday speech. A speed limit that applies on a highway is a speed that motorists may not exceed. The meaning of “limit” in mathematics is similar to that in the following sentence: “In the minute to win it competition the contestant was pushed to the limit of his abilities.” Thus, a “limit” in mathematics is something (like a number or geometrical figure) that is approached and might or might not be reached.

  It is in keeping with the historical approach of this book to begin with the method of exhaustion as an example of an occurrence of a limit in mathematics, as this is the method that Archimedes and other Greek mathematicians of his time used to calculate approximate values of certain areas, for example, the area of a disk. In section 7.3 , the concept of a limit is explained carefully using number sequences without giving the completely rigorous treatment (involving ε arguments) that are given in more advanced textbooks. Students of this book will probably not benefit from such a theoretical approach at this stage.

  The notion of the left or right limit of a function, introduced simplistically (by reading from a graph) in section 7.4 , leads to the definition of continuity of a function in section 7.5 . The property of continuity is important because many theorems about functions, for example, the Intermediate Value Theorem (in section 7.7 ), apply only to functions that are continuous.

  Most of the skills that students need to learn in this chapter are introduced in sections 7.6 and 7.8

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  eBook - PDF

  Introduction to Real Analysis

  • 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.

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  eBook - ePub

  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans, C. Evans(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Limits, continuity and differentiation 4

  Now that we have acquired the basic algebraic and geometrical tools that we need, we can begin to develop the calculus.

  After completing this chapter you should be able to □  Evaluate simple limits using the laws of limits; □  Decide, in simple cases, whether a function is continuous or not; □  Perform the processes of elementary differentiation; □  Obtain higher-order derivatives of a product using Leibniz’s theorem; □  Apply differentiation to calculate rates of change. At the end of this chapter we shall solve practical problems concerning cylinder pressure and the seepage of water into soil. 4.1      LIMITS

  One of the most important concepts in mathematics and therefore in its applications is that of a limit. We are often concerned with the long-term effects of things, or with what is likely to happen at a point of crisis – profitability, state of health, buoyancy, or stability of a structure. Such considerations often involve a limiting process.

  We shall meet this idea in several ways. In the first instance we consider the limit of a function at a point. Suppose y = f(x) has the property that f(x) can be arbitrarily close to l just by choosing x (≠ a) sufficiently close to a. If so, we say that f(x) tends to a limit l as x tends to a, and write

  f

  ( x )

    →   l       as     x   →   a

  Alternatively we say that f has a limit l at a, and write

  l     =  

  lim

  x → a

    f

  ( x )

  We do not insist that f(a) is defined, or, if it is defined, that its value shall be equal to l. In other words, the point a need not be in the domain of the function and, even if it is, the value at the point a need not be l. Indeed we are not interested at all in what happens at x = a; we are interested only in what happens when x is near a.

  □  Suppose

  y = f

  ( x )

  =

  x 2

  − 3 x + 2

  x − 1

  then the domain of this function consists of all real numbers other than x 0 1; so f(x) is not defined when x = 1. On the other hand, when x ≠ 1 we may simplify the expression for y

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 7a The Concept of Limit of a Function 7a.1 Introduction

  Addition, subtraction, multiplication, division, raising to a power, extracting a root, taking a logarithm, or a modulus are operations of elementary mathematics. In order to pass from elementary mathematics to higher mathematics, we must add to this list one more mathematical operation, namely, “finding the limit of a function ”.

  The notion of limit is an important new idea that lies at the foundation of Calculus. In fact, we might define Calculus as the study of limits. It is, therefore, important that we have a deep understanding of this concept. Although the topic of limit is rather theoretical in nature, we shall try to represent it in a very simple and concrete way.

  7a.2 Useful Notations Our work for understanding the concept of limit will be simplified if we use certain notations. Therefore, let us first get familiar with these notations:

  • Meaning of the notation x → a:
    Let x be a variable and “a ” be a constant. If x assumes values nearer and nearer to “a ” (without assuming the value “a ” itself), then we say x tends to a (or x approaches a ) and we write x → a . In other words, the procedure of giving values to x (from the domain of “f ”) nearer and nearer to “a ”, but not permitting x to assume the value “a ”, is denoted by the symbol “x → a ”.
    Thus, x → 1 means, we assign values to x which are nearer and nearer to 1 (but not permitting x to assume the value 1), which means that x comes closer and closer to “1”, reducing the distance between “x ” and “1”, in the process.
    Thus, by the statement “x ” tends to “a ”, we mean that:
    i. x ≠ a ,

    ii. x assumes values nearer and nearer to a , and

    iii. The way in which x should approach a is not specified.

    (Different ways of approaching “a ” are given below.)
  • Meaning of x → a −
    If we consider x to be approaching closer and closer to “a ” from the left side (i.e., through the values less than “a ”), then we denote this procedure by writing x →a − and read it as “x” tends to “a minus

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  eBook - ePub

  Calculus

  A Modern Approach

  • Karl Menger(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  f, whereby the law would assume the

       Third Form. provided that g and h exist.

       This statement is a restricted rule by which f can actually be computed, namely, in the cases where f is the sum of two functions whose limits at a are known. In these cases, the rule stipulates that one should simply add the known limits.

       One might, of course, introduce b, c, f, and a fourth name, and give the law the fourth and fifth forms:

       Some persons grasp a law such as the one just discussed more easily if for certain objects names are introduced that could be dispensed with, whereas others prefer formulations without superfluous names and symbols. One point, however, ought to be clear to every student regardless of his personal preference in the matter: the equivalence of the five preceding formulations of one and the same law. In fact, the recognition of this equivalence is an important step on the way to an understanding of many mathematical laws.

       For instance, from it follows that From and it follows that . Of course, these results can also be obtained, without reference to the Law about the Limits of Sums, by direct application of the limit concept to the sum function.

       It should be noted that g + h may have a limit at a, although neither g nor h has. For instance, neither sgn nor 1 − sgn has a limit at 0. Yet their sum (which is the constant function 1) has the limit 1 at 0.

       If both g and h are continuous at a, then lim g = g a and lim h = h a. Hence, by the above rules, ( g + h ) = g a + h a, which is ( g + h)a. Since, at a, a the limit of g + h and the value of g + h are equal, the function g + h is continuous at a. Similarly, g · h is continuous at a. In other words, if both g and h are continuous at a, then so are their sum and their product.

       Next, suppose that lim h = b and g = c. This means1 that, if

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  The Fundamentals of Mathematical Analysis

  • 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.

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  eBook - ePub

  Mathematical Methods for Life Sciences

  • Cinzia Bisi, Rita Fioresi(Authors)
  • 2024(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  2 Limits and Derivatives

  DOI: 10.1201/9781003343288-2

  2.1 Limits

  The concept of limit represents the foundation of differential and integral calculus: it tells us how the value f(x) of a function f changes as the variable x approaches a value x0 .

  For example, consider the linear function f(x) = x + 3. We see that f(x) approximates the number 5, as x approaches the value x0 = 2. We can easily see it, if we substitute the value 2 for x: f(2) = 2 + 3 = 5 and we look at the graph of the line y = x + 3. Indeed, for values of x close to the point x0 = 2 on the x-axis, the corresponding values of the function f are very close to f(2) = 5.

  x	f(x)
  2	5
  2.1	5.1
  1.9	4.9

  As we shall see, we need the definition of limit to understand the values of the function f at x “near” x0 , when we cannot evaluate the function f at x0 .

  Definition 2.1.1 Let f : D → ℝ and suppose that f is defined at all points of an open interval containing x0 , but not necessarily in x0 . We say that the real number L is the limit of f(x) for x approaching x0 , if, for every ε > 0, there exists δ > 0 such that, if 0 < |x − x0 | < δ, we have:

  | f ( x ) − L | < ϵ

  We write in formulas:

  lim

  x →

  x 0

  f ( x ) = L

  Let us see an example.

  Example 2.1.2 We want to check that the limit of the function f(x) = x + 3, for x approaching 2, is equal to 5, as we intuitively anticipated at the beginning of this section:

  lim

  x → 2

  x + 3 = 5

  We must show that, for any number ε > 0, we can always find a number δ > 0, such that if 0 < |x − 2| < δ, then |x + 3 − 5| < ε. Let us start by examining the last inequality:

  | x − 2 | < ϵ ⇔ 2 − ϵ < x < 2 + ϵ

  So if δ = ε, we have 0 < |x − 2| < δ.

  We invite the reader to check by exercise, using the reasoning of the previous example, that:

  lim

  x →

  x 0

  x =

  x 0

  ,       

  lim

  x →

  x 0

  k = k

  (2.1)

  where k ∈ ℝ is a constant.

  We can also define the right-hand limit and the left-hand limit of a function at x0 .

  • Right-hand limit: We define
    lim x

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  eBook - ePub

  A Concept of Limits

  • Donald W. Hight(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  . A more general generalized limit definition can be formed to include even such “limits” as “infinity.” However, we shall not undertake such a task. If you have proceeded successfully this far in this book you should be quite capable of reading more advanced texts for a more general, more advanced point of view. (See modern texts in calculus, topology, and functions of a real or complex variable.) Also, if you have been a participating reader, you may be able to form extensions of the limit theory that we have discussed to include limit infinity, limits of geometrical configurations, limiting positions of secant lines of a circle drawn through a common point, and many other notions that might be original to you. The following exercises may disclose some possible extensions of the limit concept that we have studied.

  Exercises

  1. Prove that f = {(x, 2x)} increases without bound at-the-right.

  2. Prove that g = {(x, (1/x2 ))}, x ≠ 0 increases without bound at 0; that is, prove that for every real number M there exists a real number δ > 0 such that (l/x2 ) > M if 0 < |x – 0| < δ.

  3. Let A and B be ends of a diameter of any circle O. Consider a set of points pn on the circle that satisfy the following properties: p1 is B; p2 bisects the arc bisects the arc in general pn bisects the arc . What is the “limiting position” of the ordered set of points pl P2 , P3 , ···, Pn , ··· ?

  4. Consider the ordered set of points pl , p2 , …, pn , … in Exercise 3 and imagine secant lines drawn through A and pn for every n; that is, consider the ordered set of secants Ap1 , Ap2 , …, Ap

  n

  , …. What is the “limiting position” of this ordered set of secant lines ?

  3–8 Overview

  We have studied in detail six types of limits; in abbreviated notation they are lim,

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