Differential and Integral Calculus Vol. I [1]

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N. PISKUNOV

differential and intégral calculus yo MIR PUBLISHERS ■M OSCOW

ABOUT THE BOOK

Textbook by the late Prof. Nikolai Piskunov, D.Sc. (Phys, and Math.), is devoted to the most important divisions of higher mathematics. This édition, revised and enlarged, is published in two volumes, the first volume dealing with the fo llo w in g topics: Number, Variable, Function, Lim it, C ontinuity of a Function, Dériva­ tive and D ifferential, Certain Theorems on Différentiable Functions, The Curvature of a Curve, Complex Numbers, Polynomial s, Functions of Several Variables, Applications of Differential Calculus to Solid Geometry, The Indéfini te Intégral, The Defini te Intégral, Géométrie and Mechanical Applications of the Definite Intégral. There are nu mérous examples and problems in each section of the course; many of them demonstrate the ties between mathematics and other sciences, making the book a useful aid for self-study. This is a textbook for higher technical schools that has gone through several éditions in Russian and also heen translated into French and Spanish.

H. C. n H C K Y H O B

ÆHOOEPEHUHAJIbHOE H HHTErPAJIbHOE HCHHCJIEHHfl

TOM I

H3HATEJlbCTBO «HAYKA» MOCKBA

N. PISKUNOV

DIFFERENTIAL AND

INTEGRAL CALCULUS VOL. I. Translatée! from the Russian by George Yankovsky

MI R PUBLISHERS MOSCOW

First published 1964 Second printing 1966 Third printing 1969 Second édition 1974 Fourth printing 1981

Ha

ühzauückom

H3bute

© Englisli translation, Mir Publishers, 1974

CONTENTS

CHAPTER 1. NUMBER. VARIABLE. FUNCTION

1.1 1.2 1.3 1.4 1.5

Real numbers. Real numbers aspoints on a number sc a le ................... The absolute value of a realn u m b er........................................................... Variables and c o n sta n ts................................................................................. The range of a variable .................................................................................. Ordered variables. Increasing and decreasing variables. Bounded variables . . ..................................................................................................... 1.6 F u n ctio n ............................................................................................................. 1.7 Ways of representing functions..................................................................... 1.8 Basic elementary functions.Elementary functions................................... 1.9 Algebraic fu n c tio n s......................................................................................... 1.10 Polar coordinate S ystem .................................... Exercises on Chapter 1 .............................................................................................

11 12 14 14 16 16 18 20 24 26 27

CHAPTER 2. LIMIT. CONTI NUI TY OF A FUNCTION

2.1 2.2 2.3 2.4 2.5

The limit of a variable. An infinitely large v a r i a b le ......................... The limit of a fu n c tio n ................................................................................. A function that approaches infinity. Bounded fu n ctio n s.................... Infinitesimals and their basic p r o p e r tie s................................................. Basic theorems on l i m i t s ............................................................................. si n v 2.6 The limit of the function ‘— — as x —* 0 ................................................. x 2.7 The number e ........................ . .................................................................... 2.8 Natural lo g a rith m s.......................................................................................... 2.9 Continuity of fu n ctio n s................................................................................. 2.10 Certain properties of continuous f u n c t io n s ............................................. 2.11 Comparing infinitesimals................................................................................. Exercises on Chapter 2 . . ......................................................................................

29 31 35 39 42 46 47 51 53 57 59 61

CHAPTER 3. DERI VATI VE AND DI FFERENTI AL

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Velocity of m o tio n ......................................................................................... The définition of a d é r iv a tiv e ...................................................................... Géométrie meaning of the d érivative......................................................... Differentiability of functions......................................................................... The dérivative of the function y = xn, n apositive integer . . . . Dérivatives of the functions y — sin y = cos x .................................... Dérivatives of: a constant, the product of a constant by a function, a sum, a product, and a q u o t i e n t ............................................................. The dérivative of a logarithmic fu n c tio n .................................................. The dérivative of a composite function......................................................

65 67 69 70 72 74 75 80 81

Contents

6

3.10 Dérivatives of the functions y = t a n x , y = c . o t x , y = \ n \ x \ . . . . 3.11 An hnplicit function and its d iffé r e n tia tio n ......................................... 3.12 Dérivatives of a power function for an arbitrary real exponent, of a General exponential function, and of a composite exponential f o n c t io n ............................................................................................................. 3.13 An inverse function and its différentiation............................................. 3.14 Inverse trigonométrie functions and their d iffé r e n tia tio n ................ 3.15 Basic différentiation form u las................................................................. 3.16 Parametric représentation of a f u n c t io n ................................................. 3.17 The équations of some curves in parametric fo rm ................................ 3.18 The dérivative of a function represented param etrically.................... 3.19 Hyperbolic f u n c t io n s ..................................................................................... 3.20 The differential................................................................................................. 3.21 The géométrie meaning of the d iffe r e n tia l............................................. 3.22 Dérivatives of different orders..................................................................... 3.23 Differentials of different o r d e r s ................................................................. 3.24 Dérivatives (of various orders) of implicit functions and of functions represented param etrically............................................................................. 3.25 The mechanical meaning of the second dérivative................................ 3.26 The équations of a tangent and of a normal. The lengths of a subtangent and a subnorm al......................................................................... 3.27 The géométrie meaning of the dérivative of the radius vector with respect to the polar a n g le ............................................................................. Exercises on Chapter 3 . . . . .............................................................................

83 85 87 89 92 96 98 99 102 104 107 111 112 114 116 118 11$ 122 124

CHAPTER 4. SOME THEOREMS ON D I F F E R E N T I A B L E FUNCTIONS

4.1 4.2 4.3 4.4

A theorem on the roots of a dérivative (Rolle’s t h e o r e m )................ The mean-value theorem (Lagrange’s th eorem )......................................... The generalized mean-value theorem (Cauchy’s th eorem ).................... The limit of a ratio of two infinitesimals (evaluating indeterminate forms of the type -^- )

133 135 136 137

4.5 The limit of a ratio of two infinitely large quantities (evaluating 00

\

indeterminate forms ofthe type — j ..........................................................

140

4 .6 Taylor’s form ula................................................................................................. 4.7 Expansion of the functions ex , sin x, and cos x in a Taylor sériés Exercises on Chapter 4 .............................................................................................

145 149 152

CHAPTER 5. I NVESTIGATI NG THE BEHAVI OUR OF FUNCTIONS

5.1 5 .2 5 .3 5.4

Statement of the problem ............................................................................. Increase and decrease of a f u n c t io n ......................................................... Maxima and minima of fu n c tio n s............................................................. Testing a différentiable function for maximum and minimum with a first d ériv a tiv e............................................................................................. 5.5 Testing a function for maximum and minimum with a second déri­ vative .................................... 5 .6 Maximum and minimum of a function on anin terval............................ 5.7 Applying the theory of maxima and mirtima of functions to the so­ lution of p r o b le m s......................................................................................... 5.8 Testing a function for maximum and minimum by means of Taylor’s formula . . . . . . ......................................................................................... 5.9 Convexity and concavitv of a curve.Points ofin fle c tio n ....................

155 156 157 164 166 170 171 173 175

Contents 5.10 A sym ptotes.......................................................................................................... 5.11 General plan for investigating functions and constructing graphs 5.12 Investigating curves represented param etrically..................................... Exercises on Chapter5 ...................................................

7 182 186 190 194

CHAPTER 6. THE CURVATURE OF A CURVE

6.1 6.2 6.3 6.4 6.5

Arc length and its dérivative......................................................................... C urvature.............................................................................................................. Calculation of cu rvatu re.................................................................................. Calculating the curvature of acurverepresented parametrically . . . Calculating the curvature of a curve given by an équation in polar coord i n a t e s .......................................................................................................... 6.6 The radius and circle of curvature. The centre of curvature. Evolute and i n v o l u t e ...................................................................................................... 6.7 The properties of an e v o lu t e .......................................................................... 6.8 Approximating the real roots of ané q u a t i o n ........................................... Exercises on Chapter6 ..............................................................................................

200 202 204 207 207 208 213 216 221

CHAPTER 7. COMPLEX NUMBERS. POLYNOMIALS

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12

Complex numbers. Basic d é f in itio n s ......................................................... Basic operations on complex n u m b e r s ..................................................... Powers and roots of complex n u m b e r s..................................................... Exponential function with complex exponent and its properties . . Euler’s formula. The exponential form of a complex number . . . Factoring a p olyn om ial.................................................................................. The multiple roots of a p o ly n o m ia l......................................................... Factoring a polynomial in the case of complex r o o t s ......................... Interpolation. Lagrange’s interpolation fo rm u la ..................................... Newton’s interpolation f o r m u la ................................................................. Numerical d ifféren tiation ............................................................................. On the best approximation of functions by polynomials. Chebyshev’s th e o r y .................................................................................................................. Exercises on Chapter7 ..............................................................................................

224 226 229 231 234 235 238 240 241 243 245 246 247

CHAPTER 8. FUNCTIONS OF SEVERAL VARIABLES

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

Définition of a function of several v a r ia b le s ......................................... 249 Géométrie représentation of a function of two variables ................ 252 Partial and total incrément of a f u n c t io n ............................................. 253 Continuity of a function of several v a r ia b le s......................................... 254 Partial dérivatives of a function of several v a r ia b le s......................... 257 A géométrie interprétation of the partial dérivatives of a function of two v a r i a b le s ............................................................................................. 259 Total incrément and total d iffe r e n tia l..................................................... 260 Approximation by total d ifferen tials......................................................... 263 Use of a differential to estimate errors in c a lc u la tio n s ..................... 264 The dérivative of a composite function. The total dérivative. The total differential of a composite fu n c tio n ................................................. 267 The dérivative of a function defined im p lic itly ..................................... 270 Partial dérivatives of higher o rd ers.......................... 273 Level s u r f a c e s .................................................................................................. 277 Directional dérivative ................................................................................. 278 G rad ien t.................................................................................................................... 281

8

Contents

8.16 Taylor’s formula for a function of two v a r ia b le s................................. 8.17 Maximum and minimum of a function of several variables . . . . 8.18 Maximum and minimum of a function of several variables related by given équations (conditional maxima and minima) .................... 8.19 Obtaining a function on the basis of experimental data by the method of least sq u a r es................................................................................. 8.20 Singular points of a curve . . Exercises on Chapter 8

284 286 293 298 302 307

CHAPTER 9. APPLI CATI ONS OF DI F F E RE NT I AL CALCULUS TO SOLID ( lEOMETR Y

9.1 The équations of a curve in s p a c e ............................................................. 9.2 The limit and dérivative of the vector function of a scalar argu­ ment. The équation of a tangent to a curve. The équation of a normal plane ..................................................................................................... 9.3 Rules for differentiating vectors (vector fu n ction s)................................ 9.4 The first and second dérivatives of a vector with respect to arclength. The curvature of a curve. The principal normal. The velocity and accélération of a point in curvilinear m o t i o n ......................................... 9.5 Osculating plane. Binormal. T o r s io n ........................................................ 9.6 The tangent plane and the normal to a s u r f a c e ..................................... Exercises on Chapter 9 .............................................................................................

311 314 320 322 330 335 338

CHAPTER 10. THE 1N DEFI N ITE INTEGRAL

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13

Antiderivative and the indefinite in té g r a l...................................... 341 Table of in té g r a is ................................................................................... 343 Some properties of the indefinite in té g r a l...................................... 345 Intégration by substitution (change of v a ria b le ).......................... 347 Intégrais of some functions containing a quadratic trinomial . . . 350 Intégration by p a r t s ............................................................................... 352 Rational fractions. Partial rational fractions and theirintégration 3.56 359 Décomposition of a rational fraction into partial fractions. . . . Intégration of rational f r a c t io n s ...................................................... 363 Intégrais of irrational f u n c t io n s ...................................................... 366 Intégrais of the form ^ R (x, Yax* + bx -f- c) d x ................................. 367 Intégration of certain classes of trigonométrie f u n c tio n s ................ 370 Intégration of certain irrational functions by means of trigonomét­ rie su b stitu tion s..................................................................................... .. 375 10.14 On functions whose intégrais cannot be expressed in terms of elcmentary fu n c tio n s......................................................................................... 377 Exercises on Chapter 1 0 ......................................................................................... 378 CHAPTER 11. THE DEFI N ITE I NTEGRAL

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

Statement of the problem. Lower and upper s u m s ............................. The definite intégral. Proof of the existence of a definite intégral Basic properties of the definite in té g r a l................................................. Evaluating a definite intégral. The Newton-Leibniz formula . . . Change of variable in the definite i n t é g r a l ............................ Intégration by p a r t s ..................................................................................... Improper intégrais ..................................................................................... Approximating definite in t é g r a i s .............................................................

387 389 399 402 407 408 411 419

Contents 11.9 Chebyshev’s fo rm u la ...........................................................- ...................... 11.10 Intégrais dépendent on a parameter. The gamma function . . . . 11.11 Intégration of a complex function of a reaj v a r ia b le ......................... Exercises on Chapter 1 1 .........................................................................................

9 424 429 433 433

CHAPTER 12. GEOMETRIE AND MECHANICAL APPLICATIONS OF THE DEFI N ITE I NTEGRAL

12.1 12.2 12.3 12.4

Computing areas in rcctangular coordinatcs ......................................... The area of a curvilinear sector in polar c o o r d in a te s ......................... The arc length of a curve ......................................................................... Computing the volume of a solid from the areas of parallel sections (volumes bv s l i c i n g ) ..................................................................................... 12.5 The volume of a solid of r é v o l u t i o n ......................................................... 12.6 The surface of a solid of r é v o l u t i o n ......................................................... 12.7 Computing work by the definite i n t é g r a l .......................... 12.8 Coordinates of the centre of g r a v i t y ......................................................... 12.9 Computing the moment of inertia of a line, a cirelc, and a cy 1ilicier by means of a definite i n t é g r a l ................................................................. Exercises on Chapter 1 2 ......................................................................................... Index ..........................................................................................................................

437 440 441 447 449 450 452 453 45b 458 465

CHAPTER

1

NUMBER. VARIABLE. FUNCTION

1.1 REAL NUMBERS. REAL NUMBERS AS POINTS ON A NUMBER SCALE

Number is one o! the basic concepts of mathematics. It originated in ancient times and has undergone expansion and generalization over the centuries. Whole numbers and fractions, both positive and négative, together vvith the number zéro are called rational numbers. Every rational number may be represented in the form of a ratio, -£■, of two integers p and q\ for example,

In particular, the integer p may be regarded as a ratio of two integers y ; for example,

Rational numbers may be represented in the form of periodic terminating or nonterminating fractions. Nurpbers represented by nonterminating, but nonperiodic, décimal fractions are called irrational numbers; such are the numbers Y 2, Y 3, 5 —Y 2, etc. The collection of ail rational and irrational numbers makes up the set of real numbers. The real numbers are ordered in magnitude; that is to say, for each pair of real numbers x and y there is one, and only one, of the following relations: x < y, x = y, x> y Real numbers may be depicted as points on a number scale. A number scale is an infinité straight line on which are chosen: (1) a certain point 0 called the origin, (2) a positive direction indicated by an arrow, and (3) a suitable unit of length. We shall usually make the number scale horizontal and take the positive direction to be from left to right. If the number xt is positive, it is depicted as a point at a distance OM1= x1 to the right of the origin 0; if the number x%

12

Ch. 1. Number. Variable. Function

is négative, it is represented by a point AL to the left of 0 at a distance OAL,■-=—x., (Fig. 1). The point 0 represents the number zéro. It is obvious that every real number is represented by a defmite point on the number scale. Two different real numbers are represented by different points on the number scale. The folloxving assertion is also true: each point on the number scale represents only one real number (rational or irrational). To summarize, ail real numbers and ail points on the number scale are in one-to-one correspondence: to each number there cor­ responds only one point, and conver­ sa , , ^ x x sely, to each point there corresponds -2-/ 12 3 only one number. This frequentlv enables us to regard "the number x” F,k;- 1 and "the point a" as, in a certain sense, équivalent expressions. We shall make wide use of this circumstance in our course. We state without proof the following important property of the set of real numbers: both rational and irrational numbers may be found between any two arbitrary real numbers. In geometrical terms, this proposition reads thus: both rational and irrational points may be found between any two arbitrary points on the number scale. In conclusion we give the following theorem, which, in a certain sense, represents a bridge between theory and practice. Theorem. Every irrational number a may be expressed, to any degree of accuracy, with the aid of rational numbers. Indeed, let the irrational number a > 0 and let it be required 1/ I l to evaluate a with an accuracy of —^for example, fQ» jôô» and so forth^. No matter what a is, it lies between two intégral numbers N and jV+1. We divide the interval between N and .Vr 1 into n parts; then a will lie somewhere between the rational numbers N-\~1 — and N A-n -lJr 1. Since their différence is equal to — ,each n 1 n M n of them expresses a to the given degree of accuracy, the former being too small and the latter, too large. Example. The irrational number Ÿ~ 2 is expressed by the rational numbers: 1.4 and 1.5 to one décimal place, 1.41 and 1.42 to two décimal places, 1.414 and 1.415 to three décimal places, etc. 1.2 THE ABSOLUTE VALUE OF A REAL NUMBER

Let us introduce a concept which we shall need later on: the absolute value of a real number.

1.2 The Absolute Value of a Real Number

13

Définition. The absolute value (or modulus), of a real number x (written |*|) is a nonnegative real number that satisfies the con­ ditions |* | = * if x ^ Cf | jcj = — * if * < 0 Examples. |2 | = 2,

| —5 | = 5,

|0 | = 0.

From the définition it follows that the relationship x ^ | x ( holds for any x. Let us examine some of the properties of absolute values. 1. The absolute value of an algebraic sum of several real number s is no greater than the sum of the absolute values of the ternis Proof. Let x + y ^ 0, then [* + 0l = * + ÿ < M + |0 | (since Let x -f y < 0, then

and * /< |f/|)

\ x + y \ = — ( * + y) = (— x) + (— y) < M + Iy\ This complétés the proof. The foregoing proof is readily extended to any number of terms. Examples. | —2 + 3 | < |—2 | + |3 | = 2 + 3 = 5 or 1 < 5, | - 3 — 5| = | —3 | + 1—5 | = 3 + 5 = 8 or 8 = 8.

2. The absolute value of a différence is no less than the différence of the absolute values of the minuend and subtrahend: \ x — y \ > \ x \ — \y\, | * | > | 0 | Proof. Let x — y = z, then x = y-{-z and from what has been proved M = i # + 2K M + |z| = l*/l + l*—y\ whence M — y\ thus completing the proof. 3. The absolute value of a product is equal to the product of the absolute values of the factors: \xyz\ = \x\ \y\ \z\ 4. The absolute value of a quotient is equal to the quotient of the absolute values of the dividend and the divisor: x_ y

1*1 \y\

The latter two properties follow directly from the définition of absolute value.

14

Ch. 1. Nutnber. Variable. Function 1.3 VARIABLES AND CONSTANTS

The numerical values of such physical quantities as time, length, area, volume, mass, velocity, pressure, température, etc. are determined by measurement. Mathematics deals with quantities divested of any spécifie content. From now on, when speaking of quantities, we shall hâve in view their numerical values. In various phenomena, the numerical values of certain quantities vary, while the numerical values of others remain fixed. For instance, in the uniform motion of a point, time and distance change, while the velocity remains constant. A variable is a quantity that takes on various numerical values. A constant is a quantity whose numerical values remain fixed. We shall use the letters x, y, z, u, . . . , etc. to designate variables, and the letters a, b, c, . . . , etc. to designate constants. Note. In mathematics, a constant is frequently regarded as a spécial case of variable whose numerical values are the same. It should be noted that when considering spécifie physical pheno­ mena it may happen that one and the same quantity in one phenomenon is a constant while in another it is a variable. For example, the velocity of uniform motion is a constant, while the velocity of uniformly accelerated motion is a variable. Quantities that hâve the same value under ail circumstances are called absolute constants. For example, the ratio of the circumference of a circle to its diameter is an absolute constant: n = 3.14159.... As we shall see throughout this course, the concept of a variable quantity is the basic concept of differential and intégral calculus. In “Dialectics of Nature”, Friedrich Engels wrote: “The turning point in mathematics was Descartes’ variable magnitude. With that came motion and hence dialectics in mathematics, and at once, too, of necessity the differential and intégral calculus.” 1.4 THE RANGE OF A VARIABLE

A variable takes on a sériés of numerical values. The collection of these values may differ dépend ing on the character of the problem. For example, the température of water heated under ordinary conditions will vary from room température (15-18°C) to the boiling point, 100°C. The variable quantity x = cosa can take on ail values from —1 to + 1 . The values of a variable are geometrically depicted as points on a number scale. For instance, the values of the variable x = cos a for ail possible values of a are depicted as the set of points of the interval from —1 to 1, including the points —1 and 1 (Fig. 2). Définition. The set of ail numerical values of a variable quantity is called the range of the variable.

1.4 The Range of a Variable

15

We shall now define the following ranges of a variable that will be frequently used later on. An interval is the set of ail numbers x# lying between the given points a and b (the end points) and is called closed or open accordingly as it does or does not include its end points. An open interval is the collection of ail numbers x lying between and excluding the given numbers a and b (a < b)\ it is denoted (a, b) or by means of the inequalities a < x < b. A closed interval is the set of ail num­ bers x lying between and including the Fig. 2 two given numbers a and b\ it is denoted [a, b] or, by means of inequali­ ties, If one of the numbers a or b (say, a) belongs to the interval, while the other does not, we hâve a partly closed (half-closed) interval, which may be given by the inequalities a ^ . x < b and is denoted [a, b). If the number b belongs to the set and a does not, we hâve the half-closed interval (a, b], which may be given by the inequalities a < x ^ . b . If the variable x assumes ail possible values greater than a, such an interval is denoted (a, +oo) and is represented by the conditional inequalities a < x < + oo. In the same way we regard the infinité intervals and half-closed infinité intervals represented by the conditional inequalities a ^ x < -f- oo, — oo < x < c, — oo
is the preceding value, and the value xk is the following value, irrespective of which one is the greater. Définition 1. A variable is called increasing if each subséquent value of it is greater than the preceding value. A variable is called decreasing if each subséquent value is less than the preceding value. Increasing variable quantities and decreasing variable quantities are called monolonically varying variables or simply monotonie quantities. Example. When the number of sides of a regulnr polygon inscribed in a eircle is doubled, the area s oî the polygon is an increasing variable. The area of a regular polygon circumscribed about a circle, when the number of sides is doubled, is a decreasing variable. It may be noted that not every variable quantity is necessarily increasing or decreasing. Thus, if a is an increasing variable over the interval [0, 2ji ], the variable x--=sina is not a monotonie quantity. It first increases from 0 to 1, then decreases from 1 to — 1, and then increases from — 1 to 0.

Définition 2. The variable a is called bounded if there exists a constant AI > 0 such that ail subséquent values of the variable, after a certain one, satisfy the condition —A f ^ x ^ A f or | a | ^ Af In other words, a variable is called bounded if it is possible to indicate an interval [—Af, AI] such that ail subséquent values of the variable, after a certain one, will belong to this interval. However, one should not think that the variable will necessarily assume ail values on the interval [—Al, Af], For example, the variable that assumes ail possible rational values on the interval [—2, 2] is bounded, and nevertheless it does not assume ail values on [—2, 2], namely, it does not take on the irrational values. 1.6 FUNCTION

In the study of natural phenomena and the solution of technical and mathematical problems, one finds it necessary to consider the variation of one quantity as dépendent on the variation of another.

1.6 Function

17

For instance, in studies of motion, the path traversed is regarded as a variable which varies with time. Here, the path traversed is a function of the time. . Let us consider another example. We know that the area of a circle, in terms of the radius, is Q = nR *. If the radius R takes on a variety of numerical values, the area Q will also assume various numerical values. Thus, the variation of one variable brings about a variation in the other. Here, the area of a circle Q is a function of the radius R. Let us formulate a définition of the con­ cept “function”. Définition 1. If to each value of a variable x (within a certain range) there corresponds one definite value of another variable y, then y isa function of x or, in functional notation, y = f(x), y — q>(x), and so forth. The variable x is called the independent variable or argument. The relation between the variables x and y is called a functional relation. The letter f in the functional notation y = f(x) indicates that some kind of operations must be performed on the value of x in order to obtain the value of y. In place of the notation y = f(x), u = •*« = i "F ~ » •••

We shall prove that this variable has unity as its limit. We hâve 2 n

Ch. 2. Limit. Continuity of a Function

30

For any e, ail subséquent values of the variable beginning with n, where -î- < e, or n > — , will satisfy the inequality \xn— 1 | < e, and the proof is

/î

^

complété. It will be noted here that the variable quantity decreases as it approaches the limit. Example 2. The variable x takes on the successive values X1 — 1

2

» X2 = 1

-^2 ♦ X 3 — 1

"23 > *4 = ^ T 24 * • • • » x n = ^

(

1) ” 2 H » • • •

This variable has a limit of unity. Indeed, 1 + (-!)"

_1_ 2"

-

1

_1_

2n

For any e, beginning with n, which satisfies the relation ^ < e, from which it follows that 1 1 logT 2" > T ’ nlog2 >logT or n > T ^ 2 ’ ail subséquent values of x will satisfy the relation \ x n— 1| < e. It will be no­ ted here that the values of the variable are greater than or less than the limit, and the variable approaches its limit by “oscillating about it”.

Note I. As was pointed out in Sec. 1.3, a constant quantity c is frequently regarded as a variable whose values are ail the same: .v —c. Obviously, the limit of a constant is equal to the constant itself, since we always hâve the inequality |.v—c| = |c —c| = 0 < e for any e. Note 2 . From the définition of a limit it follows that a vari­ able cannot hâve two 1imits. Indeed, if lim x = a and limx = = b (a < b ), then x must satisfy, at one and the same time, two inequalities: \x —a | < e and \x — b | < e for an arbitrarily small e; but this is impossible if e < - ~^- a(Fig. 29).

Fig. 29

Fig. 30

Note 3. One should not think that every variable has a limit. Let the variable * take on the following successive values (Fig. 30): .

_ j_

A'i — g »

« j_

.

j_

X2— 1 ^ » X3 --- g »

• • •*

- 1 _L

* 2*ifc* '^2/f+i

22*+1

2.2 The Litnit of a Function

31

For k sufficiently large, the value xik and ail subséquent values with even labels will differ from unity by as small a number as we please, while the next value x*k+l and ail subséquent values of x with odd labels will differ from zéro by as small a number as we please. Consequently, the variable x does not approach a limit. In the définition of a limit it is stated that if the variable approaches the limit a, then a is a constant. But the word “appro­ aches” is used also to describe another type of variation of a variable, as will be seen from the following définition. Définition 2 . A variable x approaches infinity if for every preassigned positive number M it is possible to indicate a value of x such that, beginning with this value, ail subséquent values of the variable satisfy the inequality |x | > M. If the variable x approaches infinity, it is called an infinitely large variable and we write x —*-- + oo and x —► —o o , the meanings of the following expressions are obvious:

2.3 A Function That Approaches Infinity

35

“/(ai) approaches b as x —*--)-oo” and “/ ( x) approaches b as x —»■—oo” or, in symbols, lim f(x) = b, X -+ + 00

lim / (ai) = b

X-* - «

2.3. A FUNCTION THAT APPROACHES INFINITY. BOUNDED FUNCTIONS

We hâve considered cases when a function f(x) approaches a certain limit b as x —*-a or as ai—►oo. Let us now take the case where the function y = f(x) approaches infinity when the argument varies in some way. Définition 1. The function f(x) approaches infinity as ai—*-a, i.e., it is an in/initely large quantity as ai—*a if for each positive number M, no matter how large, it is possible to find n 6 > 0 such that for ail values of x different from a and satisfying the condition |jc—a |< ô , we hâve the inequality \ f ( x ) \ > M . If /(ai) approaches infinity as ai —*a, we Write lim / (ai) = oo

x -*■a

or f ( x ) -*■ oo as ai—*a. If / ( ai) approaches infinity as ai—►a and, in the process, assumes only positive or only négative values, the appropriate notation is lim / (ai) = + oo or lim / (ai) = — oo. x ■* a

x ~+a

Example 1. We shall prove that

lim

7 7

— -^ = + o o . Indeed, for any M > 0

x -► 1 U — x r

wc hâve

1

>M

Ch. 2. Limit. Continuity of a Function

36 provided

The function ■^ Example M >

0

2

1

2

assumes only positive values (Fig. 34).

. We shall

prove that lim

^

^

= oo.

Indeed,

for

any

we hâve

provided 1*1 = 1*—0 *or x
0 (Fig. 35).

If the function f(x) approaches infinity as x —►oo, we wrlte lim f (x) — oo X -*■ 00

and we may hâve the particular cases lim f(x) = oo, a:

-►+ oo

lim f(x) = oo, a: -*•

- oo

lim f(x) = — oo * -*■ + oo

For example, lim X -*■ 00

= + oo,

üm x3= — oo and the like. X - * - 00

Note 1. The function y = f(x) may not approach a finite limit or infinity as x —*a or as x —►oo.

2.3 A Function That Approaches Infinity

37

Example 3. The function y = sin x defined on the infinité interval — oo < < x < + co, does not approach either a finite limit or infinity as x —*- + oo (Fig. 36).

yit

y=sLnx

0 Fig. 36 Example 4. The function y = sin -i-

defined for ail values of x, except

x = 0 , does not approach either a finite limit or infinity as x —1- 0 . The graph of this function is shown in Fig. 37.

Définition 2 . A function y = f(x) is called bounded in a given range of the argument x if there exists a positive number M such that for ail values of x in the range under considération the inequality | / (*) | 0 such that for ail values of x satisfying the inequality |x |> A f, the function f(x) is bounded. The boundedness of a function approaching a limit is decided by the following theorem. Theorem 1. If lim / (x) = b, where b is a finite number, the x -*■ a

function f(x) is bounded as x —>-a. Proof. From the équation lim f(x) = b it follows that for any x

e>

0

there will be a

6

a

such that in the neighbourhood a — Ô
-oo is unbounded because, for any M > 0, values of x can be found such that |x s in x |> A f . But the function # = xsinx is not infinitely large because it becomes zéro when x = 0, n , 2n, . . . . The graph of the function i/ = xsinx is shown in Fig. 38. Theorem 2. If lim /(x) = 6=^0, then the function y = r r \ o, x -* a i \x ) bounded function as x —>-a. Proof. From the statement of the theorem it follows that for an arbitrary e > 0 in a certain neighbourhood of the point x = a we will hâve | / (x)—b \ < e, or |/ ( x ) | —16|| < e, or —e < |/ ( x ) | — — |b |< e , or \b\ — e < |/ ( x ) < | h | + e. From the latier inequality it follows that

2.4

Infinitesimals and Their Basic Properties

For example, taking e = ^ | ô | ,

39

we get

_I2_ > _J_> _L2_

9 Ib I ^ l / W I ^ 11|*|

which means that the function -ft-t is bounded. 2.4

INFINITESIMALS AND THEIR BASIC PROPERTIES

In this section we shall consider functions approaching zéro as the argument varies in a certain manner. Définition. The function a = a(x) is called infinitésimal as x —►a or as x —>-oo if lim a(x) = 0 or lim a(x) = 0 . x -+ a

x -► oo

From the définition of a limit it follows that if, for example, lim a(x) = 0 , this means that for any preassigned arbitrarily small x -* a

positive e there will be a ô > 0 such that for ail x satisfying the condition | x — a | < ô, the condition | a (x) | < e will be satisfied.

Example 1. The function a = (x — l ) 2 is an infinitésimal as x —*-l because lim a = lim (x— 1 ) 2 = 0 (Fig. 39). X -> 1

X

Example

1 2

. The function a = — is an infinitésimal as x —►oo (Fig. 40)

(see Example 3, Sec. 2.2).

Let us establish a relationship that will be important later on. Theorem 1. I f the function y = f(x) is in the form of a sum of a constant b and an infinitésimal a: y = b + cc ( 1) then lim y —b (as x —^ a or x —-a (or as x —►oo) and does not become zéro, then H=r ~ approaches infinity. Proof. For any M > 0, no matter how large, the inequality y^y > Af will be fulfilled provided the inequality | a |< ÿ ^- is ful­ filled. The latter inequality will be fulfilled for ail values of a, from a certain one onwards, since a (x )—*0. Theorem 3. The algebraic sum of two, three or, in general, a definite number of infinitesimals is an infinitésimal function. Proof. We shall prove the theorem for two terms, since the proof is similar for any number of terms. Let u(.x:)=a(x) + p(x), where lim a(x) = 0, limP(x) = 0. We x

a.

x -*■a

shall prove that for any e > 0 , no matter how small, there will be a 6 > 0 such that when the inequality |x —a | < 6 is satisfied, the inequality | « | < e will be fulfilled. Since a(x) is an infinités­ imal, a ô, will be found such that in a neighbourhood with centre at the point a and radius àx we will hâve |a (* )| < y

2.4

Infinitésimal s and Their Basic Properties

41

Since P(x) is an infinitésimal, there will be a ô2 such that in a neighbourhood with centre at the point a and radius ô2 we will hâve | p (x) | < - j . Let us take 6 equal to the smaller o! the two quantities 6 Xand ô2; then the inequalities | a | < ^ and |P |< - |- will be fulfilled in a neighbourhood of the point a of radius ô. Hence, in this neigh­ bourhood we will hâve I« I = Ia (*) + P (*) | < I a (*) 1+ 1 P (*) I < - j + y = e and so | u | < e, as required. The proof is similar for the case when lim a(x) = 0 , lim P(x) = 0 X - * ÛD

X -► CD

Note. Later on we will hâve to consider sums of infinitesimals such that the number of terms increases with a decrease in each term. In this case, the theorem may not hold. To take an example, consider « = -^- + “ + •••+-£- where x takes on only positive x te rm s

intégral values (x = 1 , 2, 3, . . . , n, . . . ) . It is obvious that as x —►oo each term is an infinitésimal, but the sum u = l is not an infinitésimal. Theorem 4. The product of the function of an infinitésimal a = a (x) by a bounded function z — z( x), as x —*a (or x —*- 0 there will be a neighbourhood of the point x = a in which the inequality |z |< A f will be satisfied. For any e > 0 there will be a neighbourhood in which the inequality | a l< -^ j will be fulfilled. The following inequality will be fulfilled in the least of these two neighbourhoods: \a z \ < I T M = b which means that az is an infinitésimal. The proof is similar for the case x —>-oo. Two corollaries follow from this theorem. Corollary 1 . If lim a = 0, lim (5 = 0, then limap = 0 because P (x) is a bounded quantity. This holds for any finite number of factors. Corollary 2 . If lim a = 0 and c = const, then limca = 0. Theorem 5. The quotient 2~(X) y obtained by dividing the infinitesimal a(x) by a function whose limit differs from zéro is an infini­ tésimal,

42

Ch. 2. Limit. Continuity of a Function

Proof. Let lim a(x) = 0, limz(x) = b=^0. By Theorem 2, Sec. 2.3, it follows that is a bounded quantity. For this reason, the fraction = a (x) ^ is a product of an infinitésimal by a boun­ ded quantity, that is, an infinitésimal. 2.5

BASIC THEOREMS ON LIM1TS

ln this section, as in the preceding one, we shall consider sets of functions that dépend on the same argument x , where x —s an infinitésimal. Hence, lim«x«2 = = a1û'2= lim«1-limu2. * Corollary. A constant factor may be taken outside the limit sign. Indeed, if lim ul = a 1, c is a constant and, consequently, lim c=c, then lim (cux) = lime-lim ux= c-lim ux, as required. Example 2 . lim 5x3 = 5 lim *3 = 5*8 = 40 x -* 2

x -* 2

Theorem 3. The limit of a quotient of two variables is equal to the quotient of the limits of these variables if the limit of the denominator is not zéro: lim — = v

lim v

i/lim o ^^ O 1

Proof. Let lim u = a, lim v = b ^ O . Then « = a + a , v = b-{- P, where a and p are infinitesimals. We write the identities u _a + a _ a . / a + « a \ __ a ., a b — fia T ~ * + f i — T ~ T ~ \ F + $ ~ T ) ~ b " ^ (fr + fi)

or u _ a , a b — fia

T - T + 6 (6 + fi) The fraction y is a constant number, while the fraction is an infinitésimal variable by virtue of Theorems 4 and 5 (Sec. 2.4), since a b—fia is an infinitésimal, while the denominator b (b + fi) has the limit b2=£ 0. Thus, lim y = y = y j y Example 3. _ lim (3 x + 5 ) 3 lim x + 5 3x+5 x - .l _ x - .i X ™l 4x — 2 ~ lim (4*— 2) 4 lim x — 2 X -+ 1

3*1+5 4*1— 2

8

2

X -► 1

Here, we made use of the already proved theorem for the limit of a fraction because the limit of the denominator differs from zéro as x -*1. If the limit of the denominator is zéro, the theorem for the limit of a fraction is not appli­ cable, and spécial considérations hâve to be invoked. Example 4. Find lim î- — I . x 2 X —2 Here the denominator and numerator approach zéro as and, consequ­ ently, Theorem 3 is inapplicable. Perform the following identical transformation: x2— 4 _(x — 2) ( x + 2 ) =x+2 x— 2 ~ x —2

Ch. 2. Limit. Continuity of a Function

44

The transformation holds for ail values of jc different from 2 . Andso, having in view the définition of a limit, we can Write v 2 __ 4

( x - 2 ) (x + 2)

lim ----- —= lim JC -

2 X

2

jc—

X •+■ 2 X

Example 5. Find lim ----- r . As

jc

x -► î x — I

2

lim (jc+ 2 ) = 4 X — 2

-+ 1 the denominator approaches zéro but

the numerator does not (it approaches unity). Thus, the limit of the reciprocal is zéro: lim (je— 1 ) 0 X — 1__________ lira f u i 0 1 lim jc X - 1 JC X — 1

Whence, by Theorem 2 of the preceding section, we hâve 11

m - * -r==oo 1 X— \

Theorem 4. If the inequalities u ^ z ^ v are fulfilled between the corresponding values of three funet ions u = u(x), z = z(x) andv = = v(x), wkere u(x) and v(x) approach one and the same limit b as x —+a (or as x —+ oo), then z = z(x) approaches the same limit as x —>a (or as jc—►oo). Proof. For definiteness we shall consider variations of the funcctions as x — From the inequalities u ^ . z ^ . v follow the ine­ qualities u —b ^ . z —b ^ . v — b it is given that lim u = b y lim v = b x —a

x —a

Consequently, for e > 0 there will be a certain neighbourhood, with centre at the point a, in which the inequality | u — b < e will be fulfilled; likewise, there will be a certain neighbournood with centre at the point a in which the inequality |u — b | < e will be fulfilled. The following inequalities will be fulfilled in the smaller of these neighbourhoods: — e < u —& < e and —e < u —b < e and thus the inequalities — e < z—b < e will be fulfilled; that is, lim z = b x —a

Theorem 5. If as x —*a (or as jc— >■oo) the function y takes on nonnegative values y ^ O and, at the same time, approaches the limit b, then b is a nonnegative number b ^ O .

45

2.5 Basic Theorems on Limits

Proof. Assume that b < 0, then | y —b\~^\b\-, that is, the diffé­ rence modulus | y —b | is greater than the positive number |6 | and, hence, does not approach zéro as x —>-a. But then y does not approach b as x —*a\ this contradicts the statement of the theorem. Thus, the assumption that b < 0 leads to a contradiction. Consequently, b^s 0. In similar fashion we can prove that if y ^ O then lim ÿ ^ O . Theorem 6. If the inequality v~ ^u holds between corresponding values of two functions u = u (x) and v = v (x) which approach limits as x —*a (or as x —- oo), then lim o ^ lim u . Proof. It is given that v —u ^ Q . Hence, by Theorem 5, lim(u—u ) ^ 0 or limo— — lim u ^ O , and so lim lim u. Example

6

. Prove that !lm simc = 0. X -* 0

From Fig. 42 jt follows that if CM = 1 , x > 0 , then j4C = sinjc, AB = x » sin x < x. Obviously, when je < 0 we will hâve | sin jc | < | jc |. By Theorems 5 and 6 , it follows, from these inequalities, that lim sin jc = jc-*■o =

Fig.42

0.

Example 7. Prove that lim sin JC -*> o

x I

sin -75- < | sin

Indeed, Example

* \

8.

Prove

*

x

jc

|. Consequently, lim sin — =

0

x -> 0

lim

th a t

x

c o s jc

=

1;

n o te

.

th a t

JC -* 0

x

cos jc =

1 —

2

sin 2 —

therefore, lim cos x = lim ( JC -*■ 0

JC

0 \

1 —

2

sin 2 * ^ = *

)

1 —

2

lim sin2-^ -= X *+ 0

1 —

0

=

1

.

*

In some investigations concerning the limits of variables, one has to solve two independent problems: (1) to prove that the limit of the variable exists and to establish the boundaries within which the limit under considération exists; (2) to calculate the limit to the necessary degree of accuracy. The first problem is sometimes solved by means of the following theorem which will be important later on. Theorem 7. If a variable v is an increasing variable, that is, each subséquent value is greater than the preceding one, and if it is bounded, that is, v < M, then this variable has the limit lim v = a, where a ^ .M . A similar assertion may be made with respect to a decreasing bounded variable quantity.

46

Ch. 2. Limit. Corttinuity of a Function

We do not give the proof of this theorem here since it is based on the theory of real numbers, which we do not consider in this text. * In the following two sections we shall dérivé the limits of two functions that find wide application in mathematics. 2.6

THE LIMIT OF THE FUNCTION

x

AS X — ►0

The function is not defined for x = 0 since the numerator and denominator of the fraction become zéro. Let us find the limit of this function as x — —^ - > cosx We derived this inequality on the assumption t h a t x > 0 ; noting that = and cos(—x) = cosx, we conclude that it holds (—X) X for x < 0 as well. But l i mc o s x =l , Iiml = l. X-+0

X->0

* The proof of this theorem is given in G. M. Fikhtengolts* Principles of

Mathematical Analysis, Vol. I, Fizmatgiz, 1960 (in Russian).

2.7 The Number e

47

Hence, the variable 2i!l£ lies between two quantities that hâve the same limit (unity). Thus by Theorçm 4 of the preceding section, lim — = 1 The graph of the function ÿ = —

is shown in Fig. 44.

Examples.

, tanx .. sin x 1 .. sin je , 1 1 1. lim ------= lim ------ • -------= l i m ------- lim ------ = 1 - —= 1. * --»K«). X COSX x_ 0 X X^ 0 c o s x 1 X -* 0 *X X 2

0 3

.

lim

sin kx

*-►0

.. . s i n k x . . . sin(Æ*) . , , = lim k — t— = kl\m . = k - l = k (^ = const). X -+ 0

X~*Q

, 1 — co s* . lim --------------

2

lim

x

sin 2 — ..2

sin Y = lîm --------- s i n ~ = l ' 0 =

x

x -*o

x

a

.

a

olx

lim Ü ü l f

x-o

P*

0

2

lim sin ax

sin a*

ax - .. sin ax .. a 4. lim -t—s—= lim -s- • .^ o S in p * P sin P* (a = const,

\" X )

(kx-*- o)

P

1

p

P*

p = const).

2.7. THE NUMBER e

Let us consider the variable

( 1 + ^)" where n is an increasing variable that takes on the values 1, 2, 3........... Theorem 1. The variable ^ 1 ~ , as n —*■ ( 1 + t ) jc> ( , +^tt)" If x —>-oo, it is obvious that n —► = , (see Example 6 , Sec. 2.9).

Theorem 1. If a and p are équivalent infinitesimals, their différ­ ence a —p is an infinitésimal of higher order than a and than p.

61

Exercises on Chapter 2

Proof. Indeed, Um

= lim ( ! _ I ) = ! - H ç ,

= j_ j = °

1

Theorem 2. I f the différence of two infinitesimals a —p is an infinitésimal of higher order than a and thon P, then a and P are équivalent infinitesimals. Proof. Let lim a ~CL^ = 0, then lim (V 1 ——) = 0 , or 1—lim — = 0, CL J CL or

1 = lim

—, i.e., a ~ p . If l i m ^ Ê = 0, then

— 1^=0,

lim |~ = 1, that is, a ~ p . Example 7. Let a = x, p = * + * * , where x —►O. The infinitesimals oc and P are équivalent, since their différence P —a = x* is an infinitésimal of higher order than oc and than p. Indeed, lim ^~~a = lim — = lim x2= 0

x-+Q

lim

a

>—oc

P

x-+0

x-»-0 x

= üm

x-+Q

x2

lim

x-+0 \ + x 2

*+**

Example 8 . For x —►oo the infinitesimals a =

x4-1

1

and p = — are équivalent

infinitesimals, since their différence oc— p = l d z J .— - = — is an infinitésimal of higher order than

a

K

x2

x x 2

and than p. The limit of the ratio of oc and P is unity:

x±l lim -^-== lim JC—►CO p

— — lim

X -* 00

JC-+0D

lim ( 1 + — ) = 1 X

x - f 00 \

X J

X

Note. If the ratio of two infinitesimals — has no limit and does 0C not approach infinity, then p and a are not comparable in the above sense. p = xsin -^ -, where x —>-0. The infinitesimals a and O | P cannot be compared because their ratio — =s=sin^ as x —►O does not apE x a m p le

9. Let

a=x,

Xv

OC

proach either a finite limit or infinity (see Example 4, Sec. 2.3). Exercises on Chapter

2

Find the indicated limits:

1.

jc-*1

x ~r *

•

Ans •

4-

2.

11m [2sin n

x — c o s x + c o t x ).

Artt.

2.

62

3.

C h. 2. L im it.

lim

+— ?

Y% + x 4 -TT. 3

a

A ns.

.

6_ .

0.

Ans.

.. lim

x -+ « >

C o n t i n u i t y o f a F u n c t io n

4 . lim

x -f~ 1 — !— .

5 . lim

Ans. 2.


>+ oo, — o»

. . 32. lim Sin X . Ans. 4. X

.0

Ans.

V~2.

35.

x —►— oo.

. 9x sin 2 — 33. lim — ’45— . * -*>0 x

lim * c o t * .

Ans.

o

lim ( 1 — 2 ) t a n . Ans. 2-1 z .. sin (a + x ) — sin (a— x) 39. lim ---- v ■ --------------2-0 X

1

Ans.

1.

JC/

Â

lim X -*■ +

lim

-■*

o Y i-

^ CQS v

4

" f sin(Hr)

«o 38.

n .Ans. 2

COS X

ns

!• 2 arcsin* lim ----- r------- .

x -»Q

dx

„ Ans.

2

.

O

.. v ta n * — s in * 1 40. lim --- =--------. Ans.-rr-. JC* 2

cos a.

2- 0

2 —oo V

n -+ \

. 34.

11

36.

n

i4n*. — . 44. lim f l + - — ) e

lim - --* ■. Ans. 1 .

x -*>o tan x

•

lim ( 14 -— ^ . Ans. e2. 42. lim ( 1—

2 -ooV

31.

t»-■

37.

41.

as

1

Ans. e.

n J

JC/

. Ans. — . 43. lim (• *