One-Sided Limits

11 Key excerpts on "One-Sided Limits"

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  Mathematics

  An Applied Approach

  • Michael Sullivan, Abe Mizrahi(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  Whether we use a numerical argument or the graph of the function f, the variable x can get closer to c in only two ways: either by approaching c from the left, through numbers less than c, or by approaching c from the right, through numbers greater than c. If we only approach c from one side, we have a one-sided limit. The notation lim x c f (x)  N Find the One-Sided Limits of a function 1 lim x c  f (x)  L ▲ 692 Chapter 12 The Limit of a Function sometimes called the left limit, read as “the limit of f (x) as x approaches c from the left equals L,” may be described by the following statement: The notation x c  is used to remind us that x is less than c. The notation sometimes called the right limit, read as “the limit of f (x) as x approaches c from the right equals R,” may be described by the following statement: The notation x c  is used to remind us that x is greater than c. Figure 6 illustrates left and right limits. The left and right limits can be used to determine whether exists. See Figure 7. lim x c f (x) y c x x L x→c – x→c + y = f (x) y = f (x) x > c lim f (x) = R x < c lim f (x) = L (b) (a) c y x R x FIGURE 6 c y x L = R lim f (x) = lim f (x) y x L c R (a) lim f (x) ≠ lim f (x) (b) x→c – x→c + x→c – x→c + FIGURE 7 As x gets closer to c, but remains less than c, the corresponding value of f (x) gets closer to L. ▲ lim x c  f (x)  R ▲ As x gets closer to c, but remains greater than c, the corresponding value of f (x) gets closer to R. ▲ One-Sided Limits; Continuous Functions 693 Collectively, the left and right limits of a function are called One-Sided Limits of the function. Suppose that and . Then exists if and only if L  R. Furthermore, if L  R, then lim x c f (x)  L(R). lim x c f (x) lim x c  f (x)  R lim x c  f (x)  L ▲ EXAMPLE 1 Finding One-Sided Limits of a Function For the function find: (a) (b) (c) Figure 8 shows the graph of f. (a) To find , we look at the values of f when x is close to 2, but less than 2.

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  6 Chapter 1 / Limits and Continuity which is graphed in Figure 1.1.12. As x approaches 0 from the right, the values of f (x) −1 1 x y y = | x| x Figure 1.1.12 approach a limit of 1 [in fact, the values of f (x) are exactly 1 for all such x], and similarly, as x approaches 0 from the left, the values of f (x) approach a limit of −1. We denote these limits by writing lim x →0 + |x| x = 1 and lim x →0 − |x| x = −1 (13) With this notation, the superscript “+” indicates a limit from the right and the superscript “−” indicates a limit from the left. This leads to the general idea of a one-sided limit. 1.1.2 One-Sided Limits (an informal view) If the values of f (x) can be made as close as we like to L by taking values of x sufficiently close to a (but greater than a), then we write lim x →a + f (x) = L (14) and if the values of f (x) can be made as close as we like to L by taking values of x sufficiently close to a (but less than a), then we write lim x →a − f (x) = L (15) Expression (14) is read “the limit of f (x) as x approaches a from the right is L” or “f (x) approaches L as x approaches a from the right.” Similarly, expression (15) is read “the limit of f (x) as x approaches a from the left is L” or “f (x) approaches L as x approaches a from the left.” As with two-sided limits, the one- sided limits in (14) and (15) can also be written as f (x) →L as x →a + and f (x) →L as x →a − respectively. THE RELATIONSHIP BETWEEN One-Sided Limits AND TWO-SIDED LIMITS In general, there is no guarantee that a function f will have a two-sided limit at a given point a; that is, the values of f (x) may not get closer and closer to any single real number L as x → a. In this case we say that lim x →a f (x) does not exist Similarly, the values of f (x) may not get closer and closer to a single real number L as x → a + or as x → a − .

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  Brief Calculus

  An Applied Approach

  • Michael Sullivan(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Whether we use a numerical argument or the graph of the nction  the variable x can get closer to c in only two ways: either by approaching c om the left, through numbers less than c, or by approaching c om the right, through numbers greater than c. If we only approach c om one side, we have a one-sided limit. The notation lim f(x) = L x,- 252 Chapter 3 The Limit of a Function FIGURE 6 FIGURE 7 sometimes called the left limit, read as "the limit of f(x) as x approaches c om the left equals L," may be described by the llowing statement: As x gets closer to c, but remains less than c, the corresponding value of f(x) gets closer to L. The notation x - c- is used to remind us that xis less than c. The notation sometimes called the right limit, read as "the limit of f(x) as x approaches c om the right equals R," may be described by the llowing statement: As x gets closer to c, but remains greater than c, the corresponding value off (x) gets closer to R. The notation x - c + is used to remind us that xis greater than c. Figure 6 illustrates left and right limits. y X xc lim/(x) = R xc + (b) X The left and right limits can be used to determine whether lim f(x) exists. See Figure 7. y lim/(x) = lim/(x) x,- xc + (a) X y lim/(x)  lim/(x) xc- xc + (b) xc X EXAMPLE 1 One-Sided Limits: Continuous Functions 253 As Figure 7(a) illustrates, lim f(x) exists and equals the common value of the left limit xc and the right limit (L = R). In Figure 7(b), we see that limf(x) does not exist and that xc L  R. T his leads us to the llowing result: Suppose that lim x) = Land Jim f(x) = R. T hen limf(x) exists if and only if xc- xc + xc L = R. Furthermore, if L = R, then limf(x) = L(=R). xc Collectively, the left and right limits of a nction are called One-Sided Limits of the nction.

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The graphs of y = 1/(x + 1) and y = x/(x + 1) have horizontal asymptotes. 1.4 Limits (Discussed More Rigorously) 31 1.4 LIMITS (DISCUSSED MORE RIGOROUSLY) In the previous sections of this chapter we focused on the discovery of values of limits, either by sampling selected x-values or by applying limit theorems that were stated without proof. Our main goal in this section is to define the notion of a limit precisely, thereby making it possible to establish limits with certainty and to prove theorems about them. This will also provide us with a deeper understanding of some of the more subtle properties of functions. MOTIVATION FOR THE DEFINITION OF A TWO-SIDED LIMIT The statement lim x →a f (x) = L can be interpreted informally to mean that we can make the value of f (x) as close as we like to the real number L by making the value of x sufficiently close to a. It is our goal to make the informal phrases “as close as we like to L” and “sufficiently close to a” mathematically precise. To do this, consider the function f graphed in Figure 1.4.1a for which f (x) → L as x → a. For visual simplicity we have drawn the graph of f to be increasing on an open interval containing a, and we have intentionally placed a hole in the graph at x = a to emphasize that f need not be defined at x = a to have a limit there. Figure 1.4.1 Next, let us choose any positive number  and ask how close x must be to a in order for the values of f (x) to be within  units of L. We can answer this geometrically by drawing horizontal lines from the points L +  and L −  on the y-axis until they meet the curve y = f (x), and then drawing vertical lines from those points on the curve to the x-axis (Figure 1.4.1b). As indicated in the figure, let x 0 and x 1 be the points where those vertical lines intersect the x-axis. Now imagine that x gets closer and closer to a (from either side).

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  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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  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Analogous results are true for left-hand limits. EXAMPLE 9 Finding a limit using a one-sided limit theorem Find lim x l 2 1 f s x d , where f s x d 5 1 1 Ï x 2 2 . Solution The graph of f s x d 5 1 1 Ï x 2 2 is sketched in Figure 5. Using a (one-sided) limit theorem, we obtain lim x l 2 1 s 1 1 Ï x 2 2 d 5 lim x l 2 1 1 1 lim x l 2 1 Ï x 2 2 5 1 1 Ï lim x l 2 1 s x 2 2 d 5 1 1 0 5 1 . Note that lim x l 2 2 f s x d and lim x l 2 f s x d do not exist since Ï x 2 2 is not a real number if x , 2 . ■ x a L y 5 f ( x ) y 5 h ( x ) y 5 g ( x ) y FIGURE 3 x y 5 x 2 y 5 2 x 2 1 x 2 y 5 x 2 sin y FIGURE 4 y x y 5 1 1 Ï x 2 2 FIGURE 5 814 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. EXAMPLE 10 Finding a limit in an application using limit theorems Let c denote the speed of light (approximately 3.0 3 10 8 m/sec, or 186,000 mi/sec). In Einstein’s theory of relativity, the Lorentz contraction formula L 5 L 0 Î 1 2 y 2 c 2 specifies the relationship between (1) the length L of an object that is moving at a velocity y with respect to an observer and (2) its length L 0 at rest (see Figure 6). The formula implies that the length of the object measured by the observer is shorter when the object is moving than when it is at rest. Find and interpret lim y l c 2 L and explain why a left-hand limit is necessary. Solution Using (one-sided) limit theorems yields lim y l c 2 L 5 lim y l c 2 S L 0 Î 1 2 y 2 c 2 D definition of L 5 L 0 lim y l c 2 Î 1 2 y 2 c 2 theorem (6) 5 L 0 Î lim y l c 2 S 1 2 y 2 c 2 D theorem (13) 5 L 0 Ï 0 5 0.

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  CounterExamples

  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 .

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  eBook - ePub

  A Concept of Limits

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

    (Publisher)

  δ, 0). This is the same argument using the graph that we made in the above solution using algebra.

  To conclude this section let us “look at both sides” of a real number b. If you reconsider the Examples to Ponder in §2–10 , you should find that some have a left-side limit at 0 and some have a right-side limit at 0, and in some cases these two limits are equal. It is this latter type that we wish to characterize by defining a “limit at zero.”

  In general, if a function f has a left-side limit at b and a right-side limit at b and

  then we define L to be the limit of f at b and write

  You probably anticipate other ways to denote this limit, such as or

  Example 5 Consider Example Ib. of §2–10 and Examples 1 and 3 of this section. Prove that |x| = 0.

  Solution: In Examples 1 and 3 we established that

  Example 6 Consider Example IIa. of §2–10 . Determine if exists.

  Solution: The function is defined on the intervals (a, 0) and (0, c) whenever a < 0 < c. If x > 0, = x + 1; and for every ε < 0, | (x + 1) –l| < ε if 0 < x < ε = δ. Hence,

  If x < 0, = –(x + 1); and for every ε > 0, |– (x + 1) – (– 1) | = | –x | = –x < ε if 0 < 0 – x < ε = δ. Hence,

  Therefore, ; hence, does not exist.

  While the definitions of the various limits at a real number b are fresh in your mind, let us return to the question, “What if x = b ?” The answer is a nonrestrictive one: “So what–it makes no difference.” A review of the definitions reveals that the condition 0 < x – b < δ or 0 < b –x < δ determines that we do not consider x = b. Therefore, the value of the function at b, and whether or not the function has a value at b have no bearing whatsoever on the existence of a left-side limit at b, right-side limit at bf or consequently a limit at b

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  An Introduction to Analysis

  0

  • Piotr Mikusi?????ski, Jan Mikusi?????ski;;;(Authors)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  x approaches 2 from the right. The following definition expresses this situation in the precise language of analysis.

  Fig. 2.6 Graph of .

  Definition 2.4.1. We say that the number l is the right-hand limit of a function f at a point a, if for every positive number ε there exists a positive number δ such that for every x ∈ (a, a + δ) we have |f(x) − l| < ε.

  If the right-hand limit of f at a is l, we write

  The arbitrary positive ε makes precise “as close as one pleases” while the δ makes precise “sufficiently close.”

  S:I don’t like this definition. It seems to be unnecessarily complicated. I understand the examples before the definition, but it is not easy to see the connection.

  T:Only beginners have this problem. Probably the difficulty arises from the use of multiple quantifiers.

  S:What are quantifiers?

  T:The expressions for every and there exists are called quantifiers. In our definition we have three quantifiers: for every, then there exists, and finally again for every. The order of these quantifiers is essential. For ε we can take any positive number, whereas the choice of δ depends on ε. Quantifiers play a very important role in the language of mathematics. Other synonyms are also used. For instance, instead of for every one can say for each, for all, or for any. Similarly, there exists can be replaced by there is, for some, or for certain. The quantifier for every is often omitted. For example, if we say for a < x < a + δ we have |f(x) − l| < ε, we actually mean for every x such that a < x < a + δ we have |f(x) − l| < ε.

  A function that has a limit at a point may behave in a more complicated manner than in the examples given above. For instance, the right-hand limit at 0 of the function

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  930 CHAPTER 13 ■ Limits: A Preview of Calculus CONCEPTS 1. Let f be a function defined on some interval 1 a, ` 2 . Then lim xS` f 1 x 2  L means that the values of f 1 x 2 can be made arbitrarily close to by taking sufficiently large. In this case the line y  L is called a of the function y  f 1 x 2 . For example, lim xS` 1 x  , and the line y  is a horizontal asymptote. 2. A sequence a 1 , a 2 , a 3 , . . . has the limit L if the nth term a n of the sequence can be made arbitrarily close to by taking n to be sufficiently . If the limit exists, we say that the sequence ; otherwise, the sequence . SKILLS 3–4 ■ Limits from a Graph (a) Use the graph of f to find the following limits. (i) lim xS` f 1 x 2 (ii) lim xS` f 1 x 2 (b) State the equations of the horizontal asymptotes. 3. 1 1 x f y 4. 1 1 y x f 5–18 ■ Limits at Infinity Find the limit. 5. lim xS` 6 x 6. lim xS` 3 x 4 7. lim xS` 2x  1 5x  1 8. lim xS` 2  3x 4x  5 9. lim xS` 4x 2  1 2  3x 2 10. lim xS` x 2  2 x 3  x  1 11. lim tS` 8t 3  t 1 2t  1 21 2t 2  1 2 12. lim rS` 4r 3  r 2 1 r  1 2 3 13. lim xS` x 4 1  x 2  x 3 14. lim tS` a 1 t  2t t  1 b 15. lim xS` a x  1 x  1  6 b 16. lim xS` a 3  x 3  x  2 b 17. lim xS` cos x 18. lim xS` sin 2 x 19–22 ■ Estimating Limits Numerically and Graphically Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. 19. lim xS` "x 2  4x 4x  1 20. lim xS` A "9x 2  x  3xB 21. lim xS` x 5 e x 22. lim xS` a 1  2 x b 3x 23–34 ■ Limits of Sequences If the sequence with the given nth term is convergent, find its limit. If it is divergent, explain why. 23. a n  1  n n  n 2 24. a n  5n n  5 25. a n  n 2 n  1 26. a n  n  1 n 3  1 27. a n  1 3 n 28. a n  1 1 2 n n 29. a n  sin1 np/ 2 2 30. a n  cos np 31. a n  3 n 2 c n1 n  1 2 2 d 32. a n  5 n a n  4 n c n1 n  1 2 2 db 33.

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  Calculus All-in-One For Dummies (+ Chapter Quizzes Online)

  • 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.

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