Mathematics
Finding Limits of Specific Functions
Finding limits of specific functions involves determining the value that a function approaches as the input approaches a certain value. This process is essential for understanding the behavior of functions near certain points and is a fundamental concept in calculus. By finding limits, mathematicians can analyze the behavior of functions and solve various problems related to rates of change and continuity.
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12 Key excerpts on "Finding Limits of Specific Functions"
eBook - PDFPrecalculus
Functions and Graphs
- Earl Swokowski, Jeffery Cole(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
789 11.1 Introduction to Limits 11.2 Definition of Limit 11.3 Techniques for Finding Limits 11.4 Limits Involving Infinity THE CONCEPT OF limit of a function f is one of the fundamental ideas that distinguishes calculus from algebra and trigonometry. In the development of calculus in the 18th century, the limit concept was treated intuitively, as is done in Section 11.1, where we regard the function value f s x d as getting close to some number L as x gets close to a number a . The flaw in using this definition is the word close . A scientist may consider a measurement as being close to an exact value L if it is within 10 –6 cm of L . A marathon runner is close to the finish line when there are 100 yards left in the race. An astronomer sometimes measures closeness in terms of light-years. Thus, to avoid ambiguities, it is necessary to formulate a definition of limit that does not contain the word close . We do so in Section 11.2, by stating what is traditionally called the e -d definition of a limit of a function . The definition is precise and applicable to every situation we wish to con-sider. Later in the chapter we discuss properties that enable us to find many limits easily, without applying the e -d definition. Limits of Functions 11 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. In calculus and its applications we are often interested in function values f s x d of a function f when x is close to a number a , but not necessarily equal to a . In fact, there are many instances where a is not in the domain of f ; that is, f s a d is undefined.
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 - 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 - PDFMathematics
An Applied Approach
- Michael Sullivan, Abe Mizrahi(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
We may describe the meaning of as follows: For all values of x approximately equal to c, with x c, the corresponding value f (x) is approximately equal to N. lim x c f (x) N lim x c f (x) N Finding Limits Using Tables and Graphs 677 A L O O K B AC K , A L O O K F O RWA R D In Chapter 10 we defined a function and many of the proper- ties that functions can have. In Chapter 11 we discussed class- es of functions and properties that the classes have. With this as background we are ready to study the limit of a function. This concept is the bridge that takes us from the mathematics of algebra and geometry to the mathematics of calculus. Calculus actually consists of two parts: the differential cal- culus, which we discuss in Chapters 13 and 14 and the integral calculus, discussed in Chapters 15 and 16. In Chapter 17 we study the calculus of functions of two or more variables. In differential calculus we introduce another property of functions, namely the derivative of a function. We shall find that the derivative opens up a way for doing many applied problems in business, economics, and social sciences. Many of these applications involve an analysis of the graph of a function. PREPARING FOR THIS SECTION Before getting started, review the following: 12.1 Finding Limits Using Tables and Graphs > Evaluating Functions (Chapter 10, Section 10.2, pp. 550 – 551) > Piecewise-defined Functions (Chapter 10, Section 10.4, pp. 583 – 585) > Library of Functions (Chapter 10, Section 10.4, pp. 579 – 583) OBJECTIVES 1 Find a limit using a table 2 Find a limit using a graph 678 Chapter 12 The Limit of a Function From Table 1, we infer that as x gets closer to 3 the value of f (x) 5x 2 gets closer to 45. That is, ◗ When choosing the values of x in a table, the number to start with and the subse- quent entries are arbitrary. However, the entries should be chosen so that the table makes it clear what the corresponding values of f are getting close to.
eBook - PDFAnton's Calculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
A 1 A 2 A 1 A 2 A 3 Figure 1.1.6 x y a b (b) x y a b (c) x y a b (a) Figure 1.1.7 DECIMALS AND LIMITS Limits also arise in the familiar context of decimals. For example, the decimal expansion This figure shows a region called the Mandelbrot Set. It illustrates how complicated a region in the plane can be and why the notion of area requires careful definition. Andreas Nilsson/Shutterstock of the fraction 1 3 is 1 3 = 0.33333 … (3) in which the dots indicate that the digit 3 repeats indefinitely. Although you may not have thought about decimals in this way, we can write (3) as 1 3 = 0.33333 … = 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + ⋯ (4) which is a sum with “infinitely many” terms. As we will discuss in more detail later, we interpret (4) to mean that the succession of finite sums 0.3, 0.3 + 0.03, 0.3 + 0.03 + 0.003, 0.3 + 0.03 + 0.003 + 0.0003, … gets closer and closer to a limiting value of 1 3 as more and more terms are included. Thus, limits even occur in the familiar context of decimal representations of real numbers. LIMITS Now that we have seen how limits arise in various ways, let us focus on the limit concept itself. The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example, let us examine the behavior of the function f (x) = x 2 − x + 1 for x-values closer and closer to 2. It is evident from the graph and table in Figure 1.1.8 that the values of f (x) get closer and closer to 3 as values of x are selected closer and closer to 2 on either the left or the right side of 2. We describe this by saying that the “limit of x 2 − x + 1 is 3 as x approaches 2 from either side,” and we write lim x →2 (x 2 − x + 1) = 3 (5)
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.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
In this chapter we study the central idea underlying calculus: the concept of a limit. Calculus is used in modeling numerous real-life phenomena, particularly situations that involve change or motion. Limits are used in finding the instantaneous rate of change of a function as well as the area of a region with curved boundary. You will learn in calculus that these two apparently different problems are closely related. In this chapter we see how limits allow us to solve both problems. In Chapter 2 we learned how to find the average rate of change of a function. For example, to find the average speed, we divide the total distance traveled by the total time. But how can we find instantaneous speed—that is, the speed at a given instant? We can’t divide the total distance by the total time because in an instant the total distance traveled is zero and the total time spent traveling is zero! But we can find the average rate of change on smaller and smaller intervals, zooming in on the instant we want. In other words, the instantaneous speed is a limit of the average speeds. In this chapter we also learn how to find areas of regions with curved sides by using the limit process. 897 Limits: A Preview of Calculus 13 13.1 Finding Limits Numerically and Graphically 13.2 Finding Limits Algebraically 13.3 Tangent Lines and Derivatives 13.4 Limits at Infinity; Limits of Sequences 13.5 Areas FOCUS ON MODELING Interpretations of Area PF-(space1)/Alamy 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.
- Mark Ryan(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
This is the main focus of this chapter. These are the interesting limit problems, the ones that likely have infinitesimal holes, and the ones that are important for differential calculus — you see more of them in Chapter 9. With these real-deal limit problems (where plugging in often gives you 0 0 ), you can try four things: your calculator, general algebraic techniques, making a limit sandwich (a special alge- braic technique), and L’Hôpital’s rule (which is covered in Chapter 18). Figuring a limit with your calculator Your calculator is a great tool for understanding limits. It can often give you a better feel for how a limit works than the algebraic techniques can. A limit problem asks you to determine what the y-value of a function is zeroing in on as the x-value approaches a particular number. With your calculator, you can actually witness the process and the result. Say you want to eval- uate the following limit: lim x x x 5 2 25 5 . The plug-and-chug method doesn’t work because plug- ging 5 into x produces the undefined result of 0 0 . Let’s solve this limit problem with a calculator. I’ll go over two basic methods. A note about calculators and other technology: With every passing year, there are more and more powerful calculators and more and more resources on the Internet that can do calculus for you. One thing that allows these technologies to do calculus is that they can handle algebra (using CAS, a Computer Algebra System). Say you input x x 3 2 5 . These technologies can FOIL that expression and give you the algebraic answer of 2 15 2 x x . A calculator like the TI-Nspire, or any other calculator with CAS, or websites like Wolfram Alpha (www.wolfram alpha.com), can actually do the above limit problem, and all sorts of more difficult calculus problems, and give you the exact numerical or algebraic answer. 144 UNIT 3 Limits Older calculator models can’t do algebra or calculus in the real, precise, algebraic way.
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- Laura R. Van Zoest(Author)
- 2003(Publication Date)
- Information Age Publishing(Publisher)
Figure 3.1. Limit Interview Problems. x f x → 2 lim ( ) x f x →−∞ lim ( ) x f x → 1 lim ( ) x f x → 5 lim ( ) x x → + 2 3 8 lim x x x → − − 2 2 2 4 lim x x x → − 0 2 1 3 lim cos x x x →∞ lim . 5 1 1 x x x → 0 3 lim sin x x x → + 0 1 1 lim ( ) Concept of Limit in a Technological Environment 47 say that the limit of a function f as x s is some number L . These state-ments were designed to pinpoint the student’s personal limit concept so that I could draw connections between the students’ solution techniques and their understanding of limit. For instance, would students who are prone to using the calculator’s trace function be inclined to hold a dynamic view of limit, as Lauten et al. (1994) suggested? Analysis First, the techniques that each student used to evaluate each limit problem were recorded. Techniques included algebraic methods, graphing, graph-ing and using the calculator’s trace function, graphing and evaluating the function at single points, generating tables of function values, and intuitive reasoning. Next, in order to assess each student’s personal concept image of limit, responses to the true and false statements about the concept (see Table 1) were considered as well as the student’s choice of a best descrip-tion of limit among these. Each student’s personal description of limit was also classified by its correspondence to the limit concept statements. These personal descriptions were coded according to the six views of limit identi-fied in Table 1 and used to confirm the student’s image of limit evidenced by the earlier questions. For example, students who described a limit as a particular value were coded as having a static view of limit. If a student described a limit as getting closer and closer to some point, then they were classified as having a dynamic view of limit. RESULTS AND DISCUSSION All students were successful on problems one through four (see Figure 3.1 for problems 1 through 10).
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
REMARK Be sure you understand the mathematical conventions regarding parentheses and trigonometric functions. For instance, in Example 10, sin 4x means sin(4x). The limit of f (x) as x approaches 0 is 1. Figure 2.24 - 2 π 2 π - 2 4 f (x) = tan x x The limit of g(x) as x approaches 0 is 4. Figure 2.25 - 2 π 2 π - 2 6 g(x) = sin 4x x 2.3 Evaluating Limits Analytically 91 2.3 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CONCEPT CHECK 1. Polynomial Function Describe how to find the limit of a polynomial function p(x) as x approaches c. 2. Indeterminate Form What is meant by an indeterminate form? 3. Squeeze Theorem In your own words, explain the Squeeze Theorem. 4. Special Limits List the three special limits. Finding a Limit In Exercises 5–18, find the limit. 5. lim xuni2192-3 (2x + 5) 6. lim xuni21929 (4x - 1) 7. lim xuni2192-3 (x 2 + 3x) 8. lim xuni21921 (2x 3 - 6x + 5) 9. lim xuni21923 radical.alt2x + 8 10. lim xuni21922 3 radical.alt212x + 3 11. lim xuni2192-4 (1 - x) 3 12. lim xuni21920 (3x - 2) 4 13. lim xuni21922 3 2x + 1 14. lim xuni2192-5 5 x + 3 15. lim xuni21921 x x 2 + 4 16. lim xuni21921 3x + 5 x + 1 17. lim xuni21927 3x radical.alt2x + 2 18. lim xuni21923 radical.alt2x + 6 x + 2 Finding Limits In Exercises 19–22, find the limits. 19. f (x) = 5 - x, g(x) = x 3 (a) lim xuni21921 f (x) (b) lim xuni21924 g(x) (c) lim xuni21921 g( f (x)) 20. f (x) = x + 7, g(x) = x 2 (a) lim xuni2192-3 f (x) (b) lim xuni21924 g(x) (c) lim xuni2192-3 g( f (x)) 21. f (x) = 4 - x 2 , g(x) = radical.alt2x + 1 (a) lim xuni21921 f (x) (b) lim xuni21923 g(x) (c) lim xuni21921 g( f (x)) 22. f (x) = 2x 2 - 3x + 1, g(x) = 3 radical.alt2x + 6 (a) lim xuni21924 f (x) (b) lim xuni219221 g(x) (c) lim xuni21924 g( f (x)) Finding a Limit of a Transcendental Function In Exercises 23–36, find the limit of the transcendental function. 23. lim xuni2192πH208622 sin x 24. lim xuni2192π tan x 25.
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.
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