Mathematics
Limits at Infinity
Limits at infinity refer to the behavior of a function as the input values approach positive or negative infinity. In mathematical terms, a limit at infinity is used to describe the long-term behavior of a function. It helps determine the value that a function approaches as the input values become increasingly large or small.
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6 Key excerpts on "Limits at Infinity"
eBook - PDFCalculus
Early Transcendental 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.
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.
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
C H A P T E R 7 LIMITS 7.1 INTRODUCTION 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. They are the algebraic skills required for calculating limits.
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
If the function f(x) tends to a finite limit A then the difference/^)— A is infinitesimal, and conversely. When f(x) | tends to + oo it is said to be an in-finitely large quantity*. Finally, it is easy to extend to the considered general case the theorems at the end of Sec. 31, establishing the relation between infinitely small and infinitely large quantities. 33. Another definition of the limit of a function. The concept of a limit of a function f(x) when x tends to a has been constructed on the basis of the more fundamental concept of limit of a sequence examined earlier. However, we can present another definition of limit of a function, without using at all the concept of a limit of a sequence. We first confine ourselves to the case when both numbers a and A are finite. Then, assuming that a is a point of condensation of the domain 9C where the function f(x) is given, the new definition of the limit is as follows: A function/(x) has the limit A when x tends to a, if for any number 6 > 0 a number δ > 0 can be found such that f(x)-A a where a is finite, it is also said that at the point a the function is infinite. X From the fact that a is a point of condensation for 9C it follows that such values of x certainly exist in the neighbourhood (a— ô,a+ <5) of a. 64 3. THEORY OF LIMITS formulated is satisfied, and according to an arbitrary ε > 0 the corresponding (in the stated meaning) number <5 > 0 has been found. Let us extract from 9C an arbitrary sequence (2) converging to a, (all x n are distinct from a). By definition of the limit of the sequence, to the number δ > 0 there corresponds a number N such that for n > N the inequality | x n — a < ô is satisfied, and consequently (see (8))|/(x„) — A | < , also.
eBook - PDFAn Introduction to Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
- Robert A. Rankin, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
CHAPTER 2 LIMITS AND CONTINUITY 7. LIMITS OF REAL FUNCTIONS DEFINED ON THE POSITIVE INTEGERS 7.1. Deflnftions and examples. In this section we are concerned with functions defined for all sufficiently large positive integers; thus f is defined on ‚( C 0 ) for some real C 0 . The results apply, in particular, to sequences. We assume throughout the section that all numbers and functions are real; later on, in § 16.3, many of the theorems and examples will be extended to complex numbers and functions. Our purpose is to assign a precise meaning to statements such as 1(n) tends to A as n tends to infinity. The definition that we give of this statement must clarify the rather vague notion of tending and dispense with the word infinity which we have not defined as a separate concept. At the same time, it must make it clear that f (n) can be made as close to A as we please by taking n large enough. The following definition satisfies these requirements. DEFINITION 7.1.1. Suppose that f is a function defined on ‚ ( C 0 ) and that A is a fixed number. Then the statement (A 1 ) 1(n) tends to the (finite) limit 4 as n tends to infinity means: ( A) For every positive e, no matter how small, there exists a real number C >_ C 0 , where C may depend upon the choice of e, such that 1(n) — A e for all integers n > C. (1) The statement ( A 1 ) is written symbolically as (A 2 ) /(n) -+ A as n -, oo , or ( A 3 ) lim f (n) = A . n -+ Co In (A 1 ), (A 2 ) and (A 3 ), the phrases and notations employing the word `infinity' and symbol oo are to be regarded as convenient abbreviations of the more precise statement (A); i.e. infinity is assigned no meaning apart from its context. 3* 35 36 AN INTRODUCTION TO MATHEMATICAL ANALYSIS Before giving examples, we make a number of general remarks about this definition, dividing them into three groups acc~rding as they refer to the numbers e, X or A.
eBook - PDFPrecalculus
Functions and Graphs
- Earl Swokowski, Jeffery Cole(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
0, there is a number N , 0 such that if x , N , then ) f s x d 2 L ) , e . SECTION 11.4 Limits Involving Infinity 821 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. Limit theorems analogous to those in Section 11.3 may be established for limits involving infinity. In particular, the theorems concerning limits of sums, products, quotients, and n th roots of functions hold if x S ` or x S 2` . We can also show that lim x l ` c 5 c and lim x l 2` c 5 c . A proof of the next theorem, using the definition of an infinite limit of a function, is given in Appendix V. The preceding theorem is useful for investigating limits of rational func-tions. Specifically, to find lim x l ` f s x d or lim x l 2` f s x d for a rational function f, first divide the numerator and denominator of f s x d by x n , where n is the high-est power of x that appears in the denominator, and then use limit theorems. This technique is illustrated in the next three examples. EXAMPLE 4 Finding an infinite limit of a rational function Find lim x l 2` 2 x 2 2 5 3 x 2 1 x 1 2 . Solution The highest power of x in the denominator is 2. Hence, by the rule stated in the preceding paragraph, we divide the numerator and the de-nominator by x 2 and then use limit theorems.
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