Mathematics
Limits
In mathematics, a limit is a fundamental concept used to describe the behavior of a function as it approaches a certain value. It represents the value that a function approaches as the input approaches a specific point. Limits are essential for understanding continuity, derivatives, and integrals in calculus.
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11 Key excerpts on "Limits"
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- 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- 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 - PDF- Corey M. Dunn(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Section 5.7: Troubleshooting Guide for Limits and Continuity. A list of frequently asked questions and their answers, arranged by section. Come here if you are struggling with any material from this chapter. 253 � 254 Introduction to Analysis 5.1 Limits of Functions We begin this chapter with the definition of a limit of a real-valued function, one of the most important tools in all of real analysis. We begin with presenting a formulation of its definition, followed by some examples. We close the section with a convenient reformulation of Limits in terms of neighborhoods, and a proof that Limits which exist are unique. We close the section with a short discussion of one-sided Limits, and a result that relates one-sided Limits to “regular” Limits. 5.1.1 Formulating the limit of a function We wish to formulate a precise definition of the limit of f ( x ) as x approaches c , and will proceed similarly to the beginning of last chapter when we began with an imprecise guiding statement, and slowly transformed it into a useable mathematical definition. The general idea of a limit of a function at a point c is to determine what value–if any–the function’s outputs are tending toward as the inputs get closer to c , but without considering what the function is actually doing at c . For now, we will only consider the situation that the proposed limit L and c itself are real numbers, and leave the infinite cases for later in this chapter, in Section 5.3. And so suppose f : D R is a function, let c be a real number, and → suppose L is the limit of f ( x ) as x approaches c . For convenience, we will write lim x c f ( x ) to be this limit. Using the motivation in boldface in the → previous paragraph, it seems reasonable to say “lim x c f ( x ) = L if and only if f ( x ) is near L whenever x is near c .” → There are two major problems with the imprecise statement above that require improvement.
eBook - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
• Find Limits of sequences. CONCEPTUAL OBJECTIVES • Understand that if a limit of a function at infinity exists, it corresponds to a horizontal asymptote. • Understand that if a limit of a sequence at infinity exists, then the sequence is convergent. If the limit does not exist, the sequence is divergent. All of the Limits we have discussed so far have been where x approaches some constant. lim x→c f (x) The result was one of two things: Either the limit existed (some real number) or the limit did not exist. Now we turn our attention to another type of limit called a limit at infinity. lim x→∞ f (x) This examines the behavior of some function ƒ as x gets large (or approaches infinity). We will also examine the Limits of sequences, a n , as n gets large, which will be useful to us in the last section when we find the area under a curve (graph of a function). 11.4.1 Limits at Infinity 11.4.1 Skill Evaluate Limits of functions at infinity. 11.4.1 Conceptual Understand that if a limit of a function at infinity exists, it corresponds to a horizontal asymptote. We actually have already found Limits at infinity in Section 2.6 when we found horizontal asymptotes. In Example 5(b) from Section 2.6, we found that the rational function f (x) = 8 x 2 + 3 _______ 4 x 2 + 1 has a horizontal asymptote y = 2, and the notation we used in that section was as x → ∞, f (x) → 2 We now use the limit notation from this chapter. Words Math The limit of f (x) as x approaches infinity is 2. lim x→∞ ( 8 x 2 + 3 _______ 4 x 2 + 1 ) = 2 The limit of f (x) as x approaches negative infinity is 2. lim x→−∞ ( 8 x 2 + 3 _______ 4 x 2 + 1 ) = 2 Challenge 59. Explain why f ′ (0) does not exist for f (x) = |x|. 60. The function f (x) = √ _ x is defined at x = 0 but the derivative f ′ (x) is not. Why? 61. Given f (x) = ax 2 + bx + c, find f ′ (x). 62. Given f (x) = a __ x 2 , find f ′ (x). 11.4 Limits at Infinity; Limits of Sequences 987 The limit laws from Section 11.2 hold for Limits at infinity.
eBook - PDF- Oscar E. Fernandez(Author)
- 2019(Publication Date)
- Princeton University Press(Publisher)
2 Limits: How to Approach Indefinitely (and Thus Never Arrive) Chapter Preview. Limits are the foundation of calculus. As stressed in our Cal-culus Workflow, they are the intermediary between finite and infinitesimal changes, the latter being the type of change calculus is all about. But there are many types of Limits—one-sided, two-sided, etc. This chapter takes you on a limit safari to teach you all about this core calculus concept. We’ll learn how to visualize, approximate, and calculate Limits; we’ll learn about the real-world applications of Limits; and along the way I’ll give you lots of tips and tricks to help you master Limits. I’ll assume you’re comfortable with the content in Appendixes A and B, so skim that first if you haven’t already. Ready? Let’s start the expedition. 2.1 One-Sided Limits: A Graphical Approach Interactive Figure Soon after we calculated the limit (1.2) (page 4) in the Zeno example from Chap-ter 1, I mentioned it was an example of a “right-hand limit.” Figure 2.1 illustrates what I mean. The rightmost graph plots Zeno’s distance traveled d versus the change d in his distance traveled. (Recall from (1.1) that d = 2 − d .) The graph excludes the point ( 0, 2 ) (hence the hole) because d = 0 (Zeno never arrives at the 2-foot mark, re-member?). But this is the static view of Zeno’s walk. Switching to a dynamics mindset produces the other three plots in the figure. As noted, observe how d approaches 2 (the arrows on the d -axis) as d approaches 0 (the arrows on the d -axis). Since ∆ d d d = 2 – ∆ d 1 2 d approaches 2 … as ∆ d approaches 0 from the right 2 1 ∆ d d 1 2 2 1 ∆ d d 1 2 2 1 ∆ d d 1 2 2 1 Figure 2.1: Illustrating the right-hand limit lim d → 0 + d = 2. 2.1 One-Sided Limits • 9 x x x x y y = f ( x ) 2 4 f ( x ) tends to 10 … as x approaches 4 from the left 10 5 10 5 10 5 10 5 y 2 4 y 2 4 y 2 4 Figure 2.2: Illustrating the left-hand limit lim x → 4 − f ( x ) = 10.
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 - 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.
- 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.
- Andrei D. Polyanin, Alexander V. Manzhirov(Authors)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
For instance, lim n →∞ (– 1 ) n n 2 = ∞ , lim n →∞ √ n = + ∞ , lim n →∞ (– n ) = – ∞ . T HEOREM (S TOLZ ). Let x n and y n be two in fi nitely large sequences, y n → + ∞ , and y n increases with the growth of n (at least for suf fi ciently large n ): y n + 1 > y n . Then lim n →∞ x n y n = lim n →∞ x n – x n – 1 y n – y n – 1 , provided that the right limit exists ( fi nite or in fi nite). Example 5. Let us fi nd the limit of the sequence z n = 1 k + 2 k + · · · + n k n k + 1 . Taking x n = 1 k + 2 k + · · · + n k and y n = n k + 1 in the Stolz theorem, we get lim n →∞ z n = lim n →∞ n k n k + 1 – ( n – 1 ) k + 1 . Since ( n – 1 ) k + 1 = n k + 1 – ( k + 1 ) n k + · · · , we have n k + 1 – ( n – 1 ) k + 1 = ( k + 1 ) n k + · · · , and therefore lim n →∞ z n = lim n →∞ n k ( k + 1 ) n k + · · · = 1 k + 1 . 240 L IMITS AND D ERIVATIVES 6.1.2-7. Upper and lower Limits of a sequence. The limit ( fi nite or in fi nite) of a subsequence of a given sequence x n is called a partial limit of x n . In the set of all partial Limits of any sequence of real numbers, there always exists the largest and the least ( fi nite or in fi nite). The largest (resp., least) partial limit of a sequence is called its upper (resp., lower ) limit . The upper and lower Limits of a sequence x n are denoted, respectively, lim n →∞ x n , lim n →∞ x n . Example 6. The upper and lower Limits of the sequence x n = (– 1 ) n are, respectively, lim n →∞ x n = 1 , lim n →∞ x n = – 1 . A sequence x n has a limit ( fi nite or in fi nite) if and only if its upper limit coincides with its lower limit: lim n →∞ x n = lim n →∞ x n = lim n →∞ x n . 6.1.3. Limit of a Function. Asymptotes 6.1.3-1. De fi nition of the limit of a function. One-sided Limits. 1 ◦ . One says that b is the limit of a function f ( x ) as x tends to a if for any ε > 0 there is δ = δ ( ε ) > 0 such that | f ( x ) – b | < ε for all x such that 0 < | x – a | < δ . Notation: lim x → a f ( x ) = b or f ( x ) → b as x → a .
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