Limits at Infinity and Asymptotes

11 Key excerpts on "Limits at Infinity and Asymptotes"

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  Calculus for The Life Sciences

  • Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  2.4 Asymptotes and Infinity In Chapter 1, we introduced the notion of infinity and represented it with the sym- bols ∞ and −∞. This symbol was used by the Romans to represent the number 1,000 (a big number to them). It was not until 1650, however, that it was first used by John Wallis (1616–1703) to represent an uncountably large number. From childhood many of us come to think of infinity as endlessness, which in a sense it is since infinity is a not a number. To mathematicians, however, infinity is a much more complicated idea than simply endlessness. As the famous mathematician David Hilbert (1862–1943) said, “The infinite! No other question has ever moved so profoundly the spirit of man” (in J. R. Newman, ed., The World of Mathematics, New York: Simon & Schuster, 1956, 1593). In this section, we tackle limits involving the infinite in two ways. First, we de- termine under what conditions functions approach a limiting value as their argu- ment becomes arbitrarily positive or arbitrarily negative. Second, we study functions whose value becomes arbitrarily large as their arguments approach a finite value where the function is not well defined. 2.4 Asymptotes and Infinity 145 Horizontal asymptotes To understand the behavior of functions as their argument becomes very positive or negative (i.e., further from the origin in either direction), we introduce horizontal asymptotes. Horizontal Asymptotes (Informal Definition) Let f be a function. We write lim x→∞ f (x) = L if f (x) can be made arbitrarily close to L for all x sufficiently large. We write lim x→−∞ f (x) = L if f (x) can be made arbitrarily close to L for all x sufficiently negative. Whenever one of these limits occurs, we say that f (x) has a horizontal asymptote at y = L . Example 1 Finding horizontal asymptotes Find the following limits involving a given function f . In each, indicate how posi- tive or negative x needs to be to ensure that f (x) is within one ten-millionth of the limiting value L .

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  Calculus

  Single Variable

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

    (Publisher)

  22 Chapter 1 / Limits and Continuity 1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with limits that describe the behavior of a function f (x) as x approaches some real number a. In this section we will be concerned with the behavior of f (x) as x increases or decreases without bound. LIMITS AT INFINITY AND HORIZONTAL ASYMPTOTES If the values of a variable x increase without bound, then we write x → +∞, and if the values of x decrease without bound, then we write x →−∞. The behavior of a function f (x) as x increases without bound or decreases without bound is sometimes called the end behavior of the function. For example, lim x →−∞ 1 x = 0 and lim x →+∞ 1 x = 0 (1–2) are illustrated numerically in Table 1.3.1 and geometrically in Figure 1.3.1. Table 1.3.1 Figure 1.3.1 In general, we will use the following notation. 1.3.1 LIMITS AT INFINITY (AN INFORMAL VIEW) If the values of f (x) eventually get as close as we like to a number L as x increases without bound, then we write lim x →+∞ f (x) = L or f (x) → L as x → +∞ (3) Similarly, if the values of f (x) eventually get as close as we like to a number L as x decreases without bound, then we write lim x →−∞ f (x) = L or f (x) → L as x →−∞ (4) Figure 1.3.2 illustrates the end behavior of a function f when lim x →+∞ f (x) = L or lim x →−∞ f (x) = L In the first case the graph of f eventually comes as close as we like to the line y = L as x Figure 1.3.2 increases without bound, and in the second case it eventually comes as close as we like to the line y = L as x decreases without bound. If either limit holds, we call the line y = L a horizontal asymptote for the graph of f . Example 1 It follows from (1) and (2) that y = 0 is a horizontal asymptote for the graph of f (x) = 1 / x in both the positive and negative directions. This is consistent with the graph of y = 1 / x shown in Figure 1.3.1.

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  Mathematics for Engineers I

  Basic Calculus

  • Gerd Baumann(Author)
  • 2010(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  If f x as x a or as x a , then the graph of y f x fall without bound and squeezes closer to the vertical line x a on the indicated side of x a . In these cases, we call the line x a a vertical asymptote. Asymptote comes from the Greek asymptotos, meaning non-intersecting. We will s oon that taking asymptote to be synonymous with no ntersecting is a bit misleading. Definition 3.1. Vertical Asymptote A line x a is called a vertical asymptote of the graph of a function f if f x or f x as x approaches a from the left or right. Example 3.5. Asymptote The four functions shown in Figure 3.9 all have a vertical asymptote at x a , which is indicated in the figure by a full or dashed line at x a . 3.2.5 Horizontal Asymptotes and Limits at Infinity Thus fa we have used limits to describe the behavior of f x as x approaches a . However, sometimes we will not be concerned with the behavior of f x near a specific x -value, but rather with how the values of f x behaves as x increases without bound or decreases without bound. This is sometimes called the end behavior of the function because it describes how the function behaves for values of x that are far from the origin. For example, it is evident from Figure 3.10 that as x increases without bound, the values of f x 1 x are positive, but get closer and closer to 0; and as x decreases without bound, the values of f x 1 x are negative, and also get closer to 0. We indicate these limiting behaviors by writing (3.15) lim x 1 x 0 and lim x 1 x 0. 156 ee s n-i r, Mathematics for En g ineers 1.5 1.0 0.5 0.0 0.5 1.0 1.5 6 4 2 0 2 4 6 x y Figure 3.10. Asymptotic behavior of the function f x 1 x . Theorem 3.3. Limits at Infinity I f the values of f x eventually get closer and closer to the number L as x increases without bound, then we write lim x f x L or f x L as x . Similarly, if the values of f x eventually get closer and closer to the number L as x decreases without bound, then we write lim x f x L or f x L as x .

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  In fact, if any rational function has a horizontal asymptote, then the right and left horizontal asymptotes must be the same. There are, however, other functions, such as exponential functions, when only one limit as x → ∞ or x → −∞ exists and the other does not. And there are functions with two different horizontal asymptotes like arctan. 8 –8 0 y 5 4 3 1 2 y = 2 ƒ(x) = 8x 2 + 3 4x 2 + 1 x x x –∞ ∞ Let ƒ be a function; then we use the following notation to represent limits at infinity: Limit at . . . Math Words Horizontal Asymptote Infinity lim x→∞ f (x) = L The limit of f (x) as x approaches infinity is L. y = L Negative Infinity lim x→−∞ f (x) = M The limit of f (x) as x approaches negative infinity is M. y = M Note the following: • Infinity (∞) and negative infinity (−∞) do not represent actual numbers, but instead indicate growth without bound. • The above limits do not have to exist. However, if they do exist, they correspond to horizontal asymptotes. Limits at Infinity Clearly, as x gets large in either the positive or the negative direction, 1 __ x approaches 0. lim x→∞ 1 _ x = 0 and lim x→−∞ 1 _ x = 0 Using these limits with the limit of a power (Limit Law 6 from Section 11.2) yields the following special limits. 988 CHAPTER 11 Limits: A Preview to Calculus Before reading Example 1, reread Section 2.6, Example 5. In Section 2.6, we stated a rule for finding horizontal asymptotes of rational functions by comparing degrees of the numerator and denominator. Here, we present an algebraic technique to determine limits at infinity for rational functions. The first step is to divide the numerator and denominator by x n , where n is the degree of the denominator. Special limits Let n be any positive integer. Then Limit at . . . Math Words Horizontal Asymptote Infinity lim x→∞ 1 __ x n = 0 The limit of 1 __ x n as x approaches infinity is 0. y = 0 Negative Infinity lim x→−∞ 1 __ x n = 0 The limit of 1 __ x n as x approaches negative infinity is 0.

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  Brief Calculus

  An Applied Approach

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

    (Publisher)

  When this limit is a number, a horizontal asymptote describes the end hehavior of the graph. For example, if Jim f(x) = N, it means that as x becomes unbounded in the positive xoo direction, the value off can be made as close as we please to N. That is, the graph of y = f(x) r x sufficiently positive is as close as we please to the horizontal line y = N. Similarly, lim f(x) = M means that the graph of y = f(x) r x sufficiently negative is x-oo as close as we please to the horizontal line y = M. These lines are called horizontal as y mptotes of the graph off See Figure 12. Horizont a l l imf(x) = M y asymptote - �� - -- - - M y=M X Horizontal N ------ ---asymptote limf(x) = N _ N x�oo Y - Finding Horizontal Asymptotes Find the horizont asymptotes, if any, of the graph of 4x 2 f(x) = x 2 + 2 SOLUTION To find any horizontal asymptotes, we need to examine two limits: Jim f(x) and Jim f(x). xoo x-oo ;� f(x) = ;� x 2 4 : 2 2 .!�  2 = 4 FIGURE 13 4 Limits at Infinity: Infinite Limits: End Behavior: Asymptotes 261 We conclude that the line y = 4 is a horizontal asymptote of the graph when x is sufficiently positive. I. f( ) 1· 4 x 2 1· 4 x 2 tm X = tm 2 + 2 = t m -2 - = 4 x -oo x -oo X x -oo X We conclude that the line y = 4 is a horizontal asymptote of the graph when x is sufficiently negative. These conclusions also explain the end behavior of the graph. t Veical Asymptotes Infinite limits are used to find vertical asymptotes. Figure 13 illustrates some of the possibilities that can occur when a nction has an infinite limit.

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

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

    (Publisher)

  - 10 - 6 - 2 2 6 10 x - 1 1 y y = f(x) = 2 2 1 1 x x - + FIGURE 7.17. Diagram for remark 7.8.4. REMARK 7.8.5. A function might have two (different) horizontal asymptotes depending on whether x approaches infinity through positive or negative values. An example is f x x x ( ) = + − 2 1 3 5 2 . Its graph is shown in figure 7.18. (It also has a vertical asymptote at x = 1 6 . .) Think about this: is it possible for a real-valued function to have three horizontal asymptotes? -10 - 2 2 10 x -1 -2 -3 1 2 3 y y = f(x) = 2 2 1 3 5 x x - - FIGURE 7.18. A function with two horizontal asymptotes. 228 • Foundations of Mathematics We will state the formal definition of a horizontal asymptote in terms of limits. The interpre- tation of the statement = →∞ f x L lim ( ) x is that, in any arbitrarily small interval I containing the value L, there will be some point on the x-axis beyond which (in the positive or negative direction) the value of the function will always be in the interval I. (This is a mathematically precise way of saying that the graph of y f x = ( ) approaches the line y L = .) DEFINITION 7.8.1. A line y L = is a horizontal asymptote of a function f x ( ) if and only if = = →∞ →-∞ f x L f x L lim ( ) or lim ( ) (or both). x x We demonstrate, by means of the examples below, how to compute a limit as x tends to infinity. The basic idea is that any fraction of the form c x k , where c is a constant and k is any natural number, tends to 0 as x tends to infinity (through positive or negative numbers). Recall that a rational function is a function of the form f x p x q x ( ) ( ) ( ) = , where p x ( ) and q x ( ) are poly- nomials. The function f x x ( ) = 1 is a basic example of a rational function. We will see that a rational function has a horizontal asymptote whenever the degree of the numerator (the highest power of x in the numerator) is the same as, or less than, the degree of the denominator (the highest power of x in the denominator).

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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)

  By letting x decrease through negative values without bound, we can make f 1 x 2 as close as we like to 1. This is expressed by writing lim xS` x 2  1 x 2  1  1 The general definition is as follows. LIMIT AT NEGATIVE INFINITY 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 L by taking x sufficiently large negative. Again, the symbol ` does not represent a number, but the expression lim xS` f 1 x 2  L is often read as “the limit of f 1 x 2 , as x approaches negative infinity, is L” The definition is illustrated in Figure 3. Notice that the graph approaches the line y  L as we look to the far left. HORIZONTAL ASYMPTOTE The line y  L is called a horizontal asymptote of the curve y  f 1 x 2 if either lim xS` f 1 x 2  L or lim xS` f 1 x 2  L 0 y=Ï y=L 0 y=Ï y=L y x x y FIGURE 3 Examples illustrating lim xS` f 1 x 2  L Copyright 2017 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. 926 CHAPTER 13 ■ Limits: A Preview of Calculus For instance, the curve illustrated in Figure 1 has the line y  1 as a horizontal as- ymptote because lim xS` x 2  1 x 2  1  1 As we discovered in Section 5.5, an example of a curve with two horizontal asymp- totes is y  tan 1 x (see Figure 4). In fact, lim xS` tan 1 x   p 2 and lim xS` tan 1 x  p 2 so both of the lines y  p/2 and y  p/2 are horizontal asymptotes. (This follows from the fact that the lines x  p/2 are vertical asymptotes of the graph of tan.) EXAMPLE 1 ■ Limits at Infinity Find lim xS` 1 x and lim xS` 1 x .

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  Mathematics

  An Applied Approach

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

    (Publisher)

  Recall that as x becomes unbounded in the positive direction or unbounded in the negative direction, the graph of a polynomial function f (x)  a n x n  a n1 x n1  . . .  a 1 x  a 0 a n  0, behaves the same as the graph of y  a n x n . In other words, as or as , we can replace by We use this fact to find lim- its of rational functions at infinity. a n x n . a n x n  a n 1 x n 1      a 1 x  a 0 x  x  f (x)  1 x 1 x lim x  1 x  0 lim x  1 x  0 f (x)  1 x EXAMPLE 1 Finding Limits at Infinity Find: (a) (b) (a) (b) We follow the same procedure as in part (a): NOW WORK PROBLEM 1. Infinite Limits Again we use the function , whose graph is given in Figure 11, to introduce the idea of infinite limits. Table 7 gives values of f for selected numbers x that are close to 0 and positive: f (x)  1 x lim x  1 x  0 lim x  5x 2  3x  2 x 3  5  lim x  5x 2 x 3  5lim x  1 x  0  lim x  3 4  3 4 lim x  3x  2 4x  1  lim x  3x 4x lim x  5x 2  3x  2 x 3  5 lim x  3x  2 4x  1 As x , 3x  2  3x and 4x  1  4x TABLE 7 x 1 0.1 0.01 0.001 0.0001 0.00001 1 10 100 1000 10,000 100,000 f (x)  1 x We see that as x gets closer to 0 from the right, the value of f (x)  is becoming unbounded in the positive direction. We express this fact by writing (3) lim x 0  1 x   1 x 1 SOLUTION 700 Chapter 12 The Limit of a Function Similarly, we use the notation (4) to indicate that as x gets closer to 0, and is negative, the values of are becoming unbounded in the negative direction. We summarize (3) and (4) by saying that f (x)  has one-sided infinite limits at 0. 1 x 1 x lim x 0  1 x   EXAMPLE 3 Finding Horizontal Asymptotes Find the horizontal asymptotes, if any, of the graph of To find any horizontal asymptotes, we need to examine two limits: and .

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  Biocalculus

  Calculus, Probability, and Statistics for the Life Sciences

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

    (Publisher)

  a . Illustrations of these four cases are given in Figure 14. (a) lim ƒ=` y 0 a x x a _ (b) lim ƒ=` a y x x a + 0 x a _ (c) lim ƒ=_` y 0 a x (d) lim ƒ=_` a y 0 x x a + FIGURE 14 (6) Definition The line x -a is called a vertical asymptote of the curve y -f s x d if at least one of the following statements is true: lim x l a f s x d -` lim x l a 2 f s x d -` lim x l a 1 f s x d -` lim x l a f s x d -2` lim x l a 2 f s x d -2` lim x l a 1 f s x d -2` For instance, the y -axis is a vertical asymptote of the curve y -1 y x 2 because lim x l 0 s 1 y x 2 d -` . In Figure 14 the line x -a is a vertical asymptote in each of the four cases shown. In general, knowledge of vertical asymptotes is very useful in sketch-ing graphs. 0 x y x=a y=ƒ a FIGURE 13 lim x l a f s x d -2` When we say a number is “large nega-tive,” we mean that it is negative but its magnitude (absolute value) is large. 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. SECTION 2.3 | Limits of Functions at Finite Numbers 121 EXAMPLE 10 | BB Anesthesiology In Example 1.2.5 we discussed the use by anesthesiologists of ventilators during surgery. We saw that C -P V where C is the steady state concentration of CO 2 in the lungs and V is the ventilation rate. As V l 0 1 , C l ` when P is constant and so lim V l 0 1 P V -` Therefore in Figure 15 the C -axis is a vertical asymptote of the graph of C . This means that as the ventilation rate becomes very low, the concentration of carbon dioxide in the lungs becomes very high.

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  Calculus

  Single Variable

  • Carl V. Lutzer, H. T. Goodwill(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  The ideas and notation are similar when t → -∞. When |t| → ∞, we some- times refer to this as the function’s end behavior, or its behavior in the far field. Returning to the idea of the falling raindrop, we write lim t→∞ v(t) = - p mg/γ . Example of limits at infinity Example 3.1. The graph of f is shown in Figure 3.1 with its horizontal asymptotes. Use it to make a conjecture about lim t→∞ f (t). Solution: The graph indicates that the number f (t) approaches to 1 as t grows, so lim t→∞ f (t) = 1. Similarly, lim t→-∞ f (t) = -1. Limits of e βt and t β as t → ∞, when β < 0 Example 3.2. Suppose β < 0. Determine (a) lim t→∞ e βt , and (b) lim t→∞ t β . Solution: We’ll use β = -2 to illustrate the solutions. Figure 3.2: The graphs of y = e -2t and y = 1/t 2 . (a) Since e -2t = ( 1 e 2 ) t is an exponential function whose base is less than 1, its value decreases toward zero as t → ∞ (see Figure 3.2), so lim t→∞ e -2t = 0. Section 2.3 Limits at Infinity 108 Figure 3.1: The graph of f for Example 3.1. (b) The function t 7-→ 1/t 2 also decreases toward zero as its argument grows, so lim t→∞ t -2 = 0. While neither of these functions ever attains a value of zero, their values get closer and closer to zero as t grows. The same thing happens with every β < 0. z Limits at Infinity Might Not Exist As with limits at finite times, when f (t) fails to approach a particular number as t → ∞, we say that the limit of f (t) at infinity does not exist or that it di- verges. There are two basic ways this could happen. When limits at infinity fail to exist because of oscillation Example 3.3. Suppose f (t) = sin(t). Explain why lim t→∞ f (t) does not exist. Solution: The number f (t) continues to oscillate between -1 and 1 forever, so it never gets closer and closer to a particular value. When a limit at infinity fails to exist because of unbounded growth Figure 3.3: The graph of g(t) from Example 3.4, along with the graphs of y = 2t/3 (low), y = t (middle), and y = 4t/3 (high).

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  Precalculus

  Functions and Graphs

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

    (Publisher)

  The line x 5 4 is a vertical asymptote for the graph. (a) If x is close to 4 and x , 4, then x 2 4 is close to 0 and negative , and lim x l 4 2 1 s x 2 4 d 3 5 2` . y x FIGURE 3 f s x d 5 1 s x 2 2 d 2 y x FIGURE 4 f s x d 5 1 s x 2 4 d 3 818 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. (b) If x is close to 4 and x . 4, then x 2 4 is a small positive number and hence l ys x 2 4 d 3 is a large positive number. Thus, lim x l 4 1 1 s x 2 4 d 3 5 ` . (c) Since the one-sided limits are not both ` or both 2` , lim x l 4 1 s x 2 4 d 3 does not exist. ■ Formulas that represent physical quantities may lead to limits involving infinity. Obviously, a physical quantity cannot approach infinity, but an analy-sis of a hypothetical situation in which that could occur may suggest uses for other related quantities. For example, consider Ohm’s law in electrical theory, which states that I 5 V y R , where R is the resistance (in ohms) of a conductor, V is the potential difference (in volts) across the conductor, and I is the current (in amperes) that flows through the conductor (see Figure 5). The resistance of certain alloys approaches zero as the temperature approaches absolute zero (approximately 2 273 8 C), and the alloy becomes a superconductor of electric-ity. If the voltage V is fixed, then, for a superconductor, lim R l 0 1 I 5 lim R l 0 1 V R 5 ` ; that is, the current increases without bound. Superconductors allow very large current to be used in generating plants or motors.

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