Calculus Min yan Department of mathematics Hong Kong University of Science and Technology November 4. 2016
Calculus Min Yan Department of Mathematics Hong Kong University of Science and Technology November 4, 2016
Contents 1.1 Limit of Sequence 1.1.1 Arithmetic Rule 9 1.1.2 Sandwich Rule 1.1.3 Some Basic Limits 19 1.2 Rigorous Definition of Sequence Limit 1.2.1 Rigorous Definition 1.2.2 The art of estimation 1.2.3 Rigorous Proof of limits 31 1. 2. 4 Rigorous Proof of Limit Propertie 1.3 Criterion for Convergence 1.3.1 Monotone Sequence 1.3.2 Application of Monotone Sequence 1.3.3 Cauchy Criterion 1.4 Infinity 1.4.1 Divergence to Infinity 1.4.2 Arithmetic Rule for Infinity 1.4.3 Unbounded Monotone Sequence 1.5 Limit of Function 1.5. 1 Properties of Function Limit 53 1.5.2 Limit of Composition Function 1. 5.3 One Sided Limit 5.4 Limit of Trigonometric Function 1.6 Rigorous Definition of Function Limit 1.6.1 Rigorous Proof of Basic Limits 67 1.6.3 Relation to Sequence Lims of Limit 1.6.2 Rigorous Proof of Propertie 1.6.4 More Properties of Function Limit 1.7 Continuity 1.7.1 Meaning of Continuity 1.7.2 Intermediate Value Theorem
Contents 1 Limit 7 1.1 Limit of Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Arithmetic Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Sandwich Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.3 Some Basic Limits . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.4 Order Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.5 Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 Rigorous Definition of Sequence Limit . . . . . . . . . . . . . . . . . . 24 1.2.1 Rigorous Definition . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.2 The Art of Estimation . . . . . . . . . . . . . . . . . . . . . . 28 1.2.3 Rigorous Proof of Limits . . . . . . . . . . . . . . . . . . . . . 31 1.2.4 Rigorous Proof of Limit Properties . . . . . . . . . . . . . . . 33 1.3 Criterion for Convergence . . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.1 Monotone Sequence . . . . . . . . . . . . . . . . . . . . . . . . 38 1.3.2 Application of Monotone Sequence . . . . . . . . . . . . . . . 42 1.3.3 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.4 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.1 Divergence to Infinity . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.2 Arithmetic Rule for Infinity . . . . . . . . . . . . . . . . . . . 50 1.4.3 Unbounded Monotone Sequence . . . . . . . . . . . . . . . . . 52 1.5 Limit of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.5.1 Properties of Function Limit . . . . . . . . . . . . . . . . . . . 53 1.5.2 Limit of Composition Function . . . . . . . . . . . . . . . . . 56 1.5.3 One Sided Limit . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.5.4 Limit of Trigonometric Function . . . . . . . . . . . . . . . . . 63 1.6 Rigorous Definition of Function Limit . . . . . . . . . . . . . . . . . . 66 1.6.1 Rigorous Proof of Basic Limits . . . . . . . . . . . . . . . . . 67 1.6.2 Rigorous Proof of Properties of Limit . . . . . . . . . . . . . . 70 1.6.3 Relation to Sequence Limit . . . . . . . . . . . . . . . . . . . 73 1.6.4 More Properties of Function Limit . . . . . . . . . . . . . . . 78 1.7 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 1.7.1 Meaning of Continuity . . . . . . . . . . . . . . . . . . . . . . 81 1.7.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 82 3
CONTENTS 1.7.3 Continuous Inverse Function 1.7.4 Continuous Change of Variable 2 Differentiation 93 2.1 Linear Approximation 2.1.1 Derivative 2.1.2 Basic derivative 2.1.3 Constant Approximation 4 One Sided Derivative 100 2.2 Property of Derivative 2.2.1 Arithmetic Combination of Linear Approximation 2.2.2 Composition of Linear Approximation 102 2.2.3 Implicit Linear Approximation 109 2.3 Application of Linear Approximation 113 2.3.1 Monotone Property and Extrema 113 2.3.2 Detect the Monotone Property 115 2.3.3 Compare Functions 118 2.3.4 First derivative Test 120 2.3.5 Optimization Problem 122 2.4 Main value theorem 125 2.4.1 Mean value Theorem 125 2.4.2 Criterion for Constant Function 127 2.4.3 L Hospital's rule 129 2.5 High Order Approximation 133 2.5.1 Taylor Expansion 2.5.2 High Order Approximation by Substitution 2.5.3 Combination of High Order Approximations 143 2.5.4 Implicit High Order Differentiation 2.5.5 Two Theoretical Examples 2.6 Application of High Order Approximation 2.6.2 Convex Function 152 ch of Graph 2.7 Numerical Application 2.7.1 Remainder Formula 2.7.2 wton's method 3 Integration 169 3.1 Area and Definite Integral 169 3.1.1 Area below Non-negative Function 3.1.2 Definite Integral of Continuous Function 3.1.3 Property of Area and Definite Integral 175 3.2 Rigorous Definition of Integral 178
4 CONTENTS 1.7.3 Continuous Inverse Function . . . . . . . . . . . . . . . . . . . 84 1.7.4 Continuous Change of Variable . . . . . . . . . . . . . . . . . 88 2 Differentiation 93 2.1 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.1.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.1.2 Basic Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.1.3 Constant Approximation . . . . . . . . . . . . . . . . . . . . . 98 2.1.4 One Sided Derivative . . . . . . . . . . . . . . . . . . . . . . . 100 2.2 Property of Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.2.1 Arithmetic Combination of Linear Approximation . . . . . . . 101 2.2.2 Composition of Linear Approximation . . . . . . . . . . . . . 102 2.2.3 Implicit Linear Approximation . . . . . . . . . . . . . . . . . . 109 2.3 Application of Linear Approximation . . . . . . . . . . . . . . . . . . 113 2.3.1 Monotone Property and Extrema . . . . . . . . . . . . . . . . 113 2.3.2 Detect the Monotone Property . . . . . . . . . . . . . . . . . 115 2.3.3 Compare Functions . . . . . . . . . . . . . . . . . . . . . . . . 118 2.3.4 First Derivative Test . . . . . . . . . . . . . . . . . . . . . . . 120 2.3.5 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 122 2.4 Main Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.4.1 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . 125 2.4.2 Criterion for Constant Function . . . . . . . . . . . . . . . . . 127 2.4.3 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.5 High Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . 133 2.5.1 Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.5.2 High Order Approximation by Substitution . . . . . . . . . . . 139 2.5.3 Combination of High Order Approximations . . . . . . . . . . 143 2.5.4 Implicit High Order Differentiation . . . . . . . . . . . . . . . 148 2.5.5 Two Theoretical Examples . . . . . . . . . . . . . . . . . . . . 149 2.6 Application of High Order Approximation . . . . . . . . . . . . . . . 151 2.6.1 High Derivative Test . . . . . . . . . . . . . . . . . . . . . . . 151 2.6.2 Convex Function . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.6.3 Sketch of Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 157 2.7 Numerical Application . . . . . . . . . . . . . . . . . . . . . . . . . . 161 2.7.1 Remainder Formula . . . . . . . . . . . . . . . . . . . . . . . . 162 2.7.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 164 3 Integration 169 3.1 Area and Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . 169 3.1.1 Area below Non-negative Function . . . . . . . . . . . . . . . 169 3.1.2 Definite Integral of Continuous Function . . . . . . . . . . . . 172 3.1.3 Property of Area and Definite Integral . . . . . . . . . . . . . 175 3.2 Rigorous Definition of Integral . . . . . . . . . . . . . . . . . . . . . . 178
CONTENTS 3.2.1 What is Area? 3.2.3 Riemann Sum 183 3.3 Numerical Calculation of Integral 3.3.1 Left and right Rule 3.3.2 Midpoint Rule and Trapezoidal Rule 186 3.3.3 Simpson's rule 189 3.4 Indefinite Integral 3.4.1 Fundamental Theorem of Calculus 3.4.2 Indefinite Integral 3.5 Properties of Integration 198 3.5.1 Linear Property 198 3.5.2 Integration by Part 204 3.5.3 Change of Variable 214 3.6 Integration of Rational Function 228 3.6.1 Rational function r+b 3.6.2 Rational Function of 3.6.3 Rational Function of sin a and cos x 3.7 Improper Integral 240 3.7.1 Definition and Property 240 3.7.2 Comparison Test 245 3.7.3 Conditional Convergence 249 3.8 Application to Geometry 3.8.1 Length of Curve 253 3.8.2 Area of Region 3. 8.3 Surface of Revolution 3.8.4 Solid of Revolution 5 Cavalieri's Principle 3.9 Polar Coordinate 3.9.1 Curves in Polar Coordinate 3.9.2 Geometry in Polar Coordinate 3.10 Application to Physics 3.10.1 Work and Pressure 3.10.2 Center of Mass 4 Series 297 4.1 Series of numbers 4.1.1 Sum of se erles 298 4. 1.2 Property of Converging Series 300 4.2 Comparison Test 4.2.1 Integral test 4.2.2 Comparison Test
CONTENTS 5 3.2.1 What is Area? . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.2.2 Darboux Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.2.3 Riemann Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.3 Numerical Calculation of Integral . . . . . . . . . . . . . . . . . . . . 184 3.3.1 Left and Right Rule . . . . . . . . . . . . . . . . . . . . . . . 184 3.3.2 Midpoint Rule and Trapezoidal Rule . . . . . . . . . . . . . . 186 3.3.3 Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.4 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.4.1 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 191 3.4.2 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.5 Properties of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.1 Linear Property . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 204 3.5.3 Change of Variable . . . . . . . . . . . . . . . . . . . . . . . . 214 3.6 Integration of Rational Function . . . . . . . . . . . . . . . . . . . . . 228 3.6.1 Rational Function . . . . . . . . . . . . . . . . . . . . . . . . . 228 3.6.2 Rational Function of n r ax + b cx + d . . . . . . . . . . . . . . . . . . 235 3.6.3 Rational Function of sin x and cos x . . . . . . . . . . . . . . . 238 3.7 Improper Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.7.1 Definition and Property . . . . . . . . . . . . . . . . . . . . . 240 3.7.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.7.3 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . 249 3.8 Application to Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.1 Length of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.2 Area of Region . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.8.3 Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . 265 3.8.4 Solid of Revolution . . . . . . . . . . . . . . . . . . . . . . . . 269 3.8.5 Cavalieri’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 276 3.9 Polar Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.9.1 Curves in Polar Coordinate . . . . . . . . . . . . . . . . . . . 284 3.9.2 Geometry in Polar Coordinate . . . . . . . . . . . . . . . . . . 288 3.10 Application to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.1 Work and Pressure . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.2 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4 Series 297 4.1 Series of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.1.1 Sum of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4.1.2 Property of Converging Series . . . . . . . . . . . . . . . . . . 300 4.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 4.2.1 Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.2.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 305
6 CONTENTS 4.2.3 Special Comparison Test 309 4.3 Conditional Convergence 4.3.1 Test for Conditional Convergence 4.3.2 Absolute v.s. Conditional 317 4.4 Power Series 320 4.4.1 Radius of Convergence 322 4.4.2 Function Defined by Power Series 325 4.5 Fourier Series 328 4.5.1 Fourier Coefficient 329 4.5.2 Complex Form of Fourier Series 4.5.3 Derivative and Integration of Fourier Series 336 4.5.4 Sum of Fourier Series 339 4.5.5 Parseval's Identity
6 CONTENTS 4.2.3 Special Comparison Test . . . . . . . . . . . . . . . . . . . . . 309 4.3 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 313 4.3.1 Test for Conditional Convergence . . . . . . . . . . . . . . . . 314 4.3.2 Absolute v.s. Conditional . . . . . . . . . . . . . . . . . . . . 317 4.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 4.4.1 Radius of Convergence . . . . . . . . . . . . . . . . . . . . . . 322 4.4.2 Function Defined by Power Series . . . . . . . . . . . . . . . . 325 4.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 4.5.1 Fourier Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 329 4.5.2 Complex Form of Fourier Series . . . . . . . . . . . . . . . . . 334 4.5.3 Derivative and Integration of Fourier Series . . . . . . . . . . . 336 4.5.4 Sum of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 339 4.5.5 Parseval’s Identity . . . . . . . . . . . . . . . . . . . . . . . . 341
Chapter 1 Limit 1.1 Limit of Sequence A sequence is an infinite list 1,2,…,n The n-th term of the sequence is an, and n is the index of the term. In this course. we will always assume that all the terms are real numbers. Here are some examples n yn=2:2,2,2 1 1 2’’n 1.-1.1 Un sin n in 1. sin 2. sin 3 Sin n Note that the index does not have to start from 1. For example, the sequence Un actually starts from n=0 (or any even integer). Moreover, a sequence does not have to be given by a formula. For example, the decimal expansions of T give a sequence n:3,3.1,3.14,3.141,3.1415,3.14159,3.141592 If n is the number of digits after the decimal point, then the sequence wn starts at n=0 Now we look at the trend of the examples above as n gets bigger. We find that n gets bigger and can become as big as we want. On the other hand, yn remains constant,zn gets smaller and can become as small as we want. This means that Un approaches 2 and zn approaches 0. Moreover, un and Un jump around and do not approach anything. Finally, wn is equal to T up to the n-th decimal place, and therefore approaches T
Chapter 1 Limit 1.1 Limit of Sequence A sequence is an infinite list x1, x2, . . . , xn, . . . . The n-th term of the sequence is xn, and n is the index of the term. In this course, we will always assume that all the terms are real numbers. Here are some examples xn = n: 1, 2, 3, . . . , n, . . . ; yn = 2: 2, 2, 2, . . . , 2, . . . ; zn = 1 n : 1, 1 2 , . . . , 1 n , . . . ; un = (−1)n : 1, −1, 1, . . . , (−1)n , . . . ; vn = sin n: sin 1, sin 2, sin 3, . . . , sin n, . . . . Note that the index does not have to start from 1. For example, the sequence vn actually starts from n = 0 (or any even integer). Moreover, a sequence does not have to be given by a formula. For example, the decimal expansions of π give a sequence wn : 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, . . . . If n is the number of digits after the decimal point, then the sequence wn starts at n = 0. Now we look at the trend of the examples above as n gets bigger. We find that xn gets bigger and can become as big as we want. On the other hand, yn remains constant, zn gets smaller and can become as small as we want. This means that yn approaches 2 and zn approaches 0. Moreover, un and vn jump around and do not approach anything. Finally, wn is equal to π up to the n-th decimal place, and therefore approaches π. 7
8 CHAPTER 1. LIMIT 日 日日 Figure 1. 1.1: Sequ Definition 1.1.1 (Intuitive). If In approaches a finite number I when n gets bigge and bigger, then we say that the sequence n converges to the limit l and write lim A sequence diverges if it does not approach a specific finite number when n gets The sequences n, Zn, Un converge respectively to 2, 0 and T. The sequences n, un, Un diverge. Since the limit describes the behavior when n gets very big, we have the following propert Proposition 1. 2. If yn is obtained from Tn by adding, deleting, or changing finitely many terms, then lim→oxn=limn→yn The equality in the proposition means that In converges if and only if yn con verges. Moreover, the two limits have equal value when both converge ample 1.. The sequence Vn+2 is obtained from Vn by deleting the first two terms. By limn→sJn =0 and Proposition 1. 1.2, we get lim, n+2 In general, we have limn-oo n+k= limn-yoo n for any integer k The example assumes limn-+oo 0, which is supposed to be intuitively obv ous. Although mathematics is inspired by intuition, a critical feature of mathematics is rigorous logic. This means that we need to be clear what basic facts are assumed in any argument. For the moment, we will always assume that we already know
8 CHAPTER 1. LIMIT n xn yn zn un vn wn Figure 1.1.1: Sequences. Definition 1.1.1 (Intuitive). If xn approaches a finite number l when n gets bigger and bigger, then we say that the sequence xn converges to the limit l and write limn→∞ xn = l. A sequence diverges if it does not approach a specific finite number when n gets bigger. The sequences yn, zn, wn converge respectively to 2, 0 and π. The sequences xn, un, vn diverge. Since the limit describes the behavior when n gets very big, we have the following property. Proposition 1.1.2. If yn is obtained from xn by adding, deleting, or changing finitely many terms, then limn→∞ xn = limn→∞ yn. The equality in the proposition means that xn converges if and only if yn converges. Moreover, the two limits have equal value when both converge. Example 1.1.1. The sequence 1 √ n + 2 is obtained from 1 √ n by deleting the first two terms. By limn→∞ 1 √ n = 0 and Proposition 1.1.2, we get limn→∞ 1 √ n = limn→∞ 1 √ n + 2 = 0. In general, we have limn→∞ xn+k = limn→∞ xn for any integer k. The example assumes limn→∞ 1 √ n = 0, which is supposed to be intuitively obvious. Although mathematics is inspired by intuition, a critical feature of mathematics is rigorous logic. This means that we need to be clear what basic facts are assumed in any argument. For the moment, we will always assume that we already know
1. 1. LIMIT OF SEQUENCE imn-ooC= c and limn-oo=0 for p>0. After the two limits are rigorously established in Examples 1.2.2 and 1.2.3, the conclusions based on the two limits become solid 1.1.1 Arithmetic rule Intuitively, if a is close to 3 and y is close to 5, then the arithmetic combinations t+y and ry are close to 3+5=8 and 3.5= 15. The intuition leads to the following property of limit Proposition1.13( Arithmetic rule). Suppose limn→xn= l and limn→∞n=k lim (an +yn)=l+k, lim can=c, lim Ingn=kl, lim En I n→0 where c is a constant and kf0 in the last equality The proposition says limn→(xn+mn)=limn→xn+limn→n. However,the equality is of different nature from the equality in Proposition 1.1.2, because the convergence of the limits on two sides are not equivalent: If the two limits on the ght converge, then the limit on the left also converges and the two sides are equal However, for In=(1)" and n =(1)n+, the limit limn-yoo(an +yn)=0 on the left converges. but both limits on the right diverge Exercise111. Explain that limn→∞n= l if and only if limn→∞(xn-l)=0. Exercise 1. 1.2. Suppose In and yn converge. Explain that limn-yoo nin=0 implies either limn-oo In=0 or limn-+oo n =0. Moreover, explain that the conclusion fails if an and n are not assumed to converge Example 1. 1.2. We have 2+ 2m2+n lim li 2+lin mn→1-limn→∞=+limn+nmn→ lin 2+0 0+0·0
1.1. LIMIT OF SEQUENCE 9 limn→∞ c = c and limn→∞ 1 np = 0 for p > 0. After the two limits are rigorously established in Examples 1.2.2 and 1.2.3, the conclusions based on the two limits become solid. 1.1.1 Arithmetic Rule Intuitively, if x is close to 3 and y is close to 5, then the arithmetic combinations x+y and xy are close to 3+5 = 8 and 3·5 = 15. The intuition leads to the following property of limit. Proposition 1.1.3 (Arithmetic Rule). Suppose limn→∞ xn = l and limn→∞ yn = k. Then limn→∞ (xn + yn) = l + k, limn→∞ cxn = cl, limn→∞ xnyn = kl, limn→∞ xn yn = l k , where c is a constant and k 6= 0 in the last equality. The proposition says limn→∞(xn + yn) = limn→∞ xn + limn→∞ yn. However, the equality is of different nature from the equality in Proposition 1.1.2, because the convergence of the limits on two sides are not equivalent: If the two limits on the right converge, then the limit on the left also converges and the two sides are equal. However, for xn = (−1)n and yn = (−1)n+1, the limit limn→∞(xn + yn) = 0 on the left converges, but both limits on the right diverge. Exercise 1.1.1. Explain that limn→∞ xn = l if and only if limn→∞(xn − l) = 0. Exercise 1.1.2. Suppose xn and yn converge. Explain that limn→∞ xnyn = 0 implies either limn→∞ xn = 0 or limn→∞ yn = 0. Moreover, explain that the conclusion fails if xn and yn are not assumed to converge. Example 1.1.2. We have limn→∞ 2n 2 + n n2 − n + 1 = limn→∞ 2 + 1 n 1 − 1 n + 1 n2 = limn→∞ 2 + 1 n limn→∞ 1 − 1 n + 1 n2 = limn→∞ 2 + limn→∞ 1 n limn→∞ 1 − limn→∞ 1 n + limn→∞ 1 n · limn→∞ 1 n = 2 + 0 1 − 0 + 0 · 0 = 2
10 CHAPTER 1. LIMIT The arithmetic rule is used in the second and third equalities. The limits limn-ooc 1 c and lim =0 are used in the fourth equalit Exercise 1.1.3. Find the limits 留++ (n+1)(n+2 2n3+3n 6 n2+3)3 (n+1)(n+2) Exercise 1.1.4. Find the limits ++ (√n+1)(√m+2) 3=切 2n-1 4n+1 (n+1)(n+ 9.( Exercise 1.1.5. Find the limits a +6 n+bvn+c (c√m+d) n (c√+d)3 n+a cn+d (a√n+b)2 m2+bn+c (√m+a)(√m+b) (cVn+d) Exercise 1.1.6. Show that 0. if lim …+a1n+a0 +bin+bo ifp= g and b≠0. bq Exercise 1.1.7. Find the limits 20 lo (2n+1)2-1 10m2-5 10n-5 Exercise 1.1. 8. Find the limits
10 CHAPTER 1. LIMIT The arithmetic rule is used in the second and third equalities. The limits limn→∞ c = c and limn→∞ 1 n = 0 are used in the fourth equality. Exercise 1.1.3. Find the limits. 1. n + 2 n − 3 . 2. n + 2 n2 − 3 . 3. 2n 2 − 3n + 2 3n2 − 4n + 1 . 4. n 3 + 4n 2 − 2 2n3 − n + 3 . 5. (n + 1)(n + 2) 2n2 − 1 . 6. 2n 2 − 1 (n + 1)(n + 2). 7. (n 2 + 1)(n + 2) (n + 1)(n2 + 2). 8. (2 − n) 3 2n3 + 3n − 1 . 9. (n 2 + 3)3 (n3 − 2)2 . Exercise 1.1.4. Find the limits. 1. √ n + 2 √ n − 3 . 2. √ n + 2 n − 3 . 3. 2 √ n − 3n + 2 3 √ n − 4n + 1 . 4. √3 n + 4√ n − 2 2 √3 n − n + 3 . 5. ( √ n + 1)(√ n + 2) 2n − 1 . 6. 2n − 1 ( √ n + 1)(√ n + 2). 7. ( √ n + 1)(n + 2) (n + 1)(√ n + 2). 8. (2 − √3 n) 3 2 √3 n + 3n − 1 . 9. ( √3 n + 3)3 ( √ n − 2)2 . Exercise 1.1.5. Find the limits. 1. n + a n + b . 2. √ n + a n + b . 3. n + a n2 + bn + c . 4. √ n + a n + b √ n + c . 5. ( √ n + a)(√ n + b) cn + d . 6. cn + d ( √ n + a)(√ n + b) . 7. an3 + b (c √ n + d) 6 . 8. (a √3 n + b) 2 (c √ n + d) 3 . 9. (a √ n + b) 2 (c √3 n + d) 3 . Exercise 1.1.6. Show that limn→∞ apn p + ap−1n p−1 + · · · + a1n + a0 bqnq + bq−1nq−1 + · · · + b1n + b0 = 0, if p < q, ap bq , if p = q and bq 6= 0. Exercise 1.1.7. Find the limits. 1. 1010n n2 − 10 . 2. 5 5 (2n + 1)2 − 1010 10n2 − 5 . 3. 5 5 (2√ n + 1)2 − 1010 10n − 5 . Exercise 1.1.8. Find the limits
1. 1. LIMIT OF SEQUENCE n+a n+c n+b n+d b n2+d n +b td 3+b +b-+d9.+b-边+d Exercise 1.1.9. Find the limits n2+ain+ao n2+cin+ 1 n+b n+d n+ n-+aln+a0 n-+Cin+ co m2+ bin+b0 n2+din+do Exercise 1.1.10. Find the limits, p, q >0 +c +d anP+bn9+c 又.n2p+a1m+a2 anq+bnp+c m2q+61n9+b 1.1.2 Sandwich rule The following property reflects the intuition that if a and z are close to 3, then anything between s and z should also be close to 3 Proposition 1.1.4 (Sandwich Rule). Suppose In Un En for sufficiently big n. If mn→xn=limn→oxn=l, then lim→oyn=l lote that something holds for sufficiently big n is the same as something fails for only finitely many n havinple 1.1.3. By 2n-3 >n for sufficiently big n(in fact, n>3 is enough),we E 0<√2n-3 Then by lin→a0=limn→ 0 and the sandwich rule, we get lim √2n-3 On the other hand, for sufficiently big n, we have n+1 2n and n-1 and therefore vn+I√2n2V2
1.1. LIMIT OF SEQUENCE 11 1. n n + 1 − n n − 1 . 2. n 2 n + 1 − n 2 n − 1 . 3. n √ n + 1 − n √ n − 1 . 4. n + a n + b − n + c n + d . 5. n 2 + a n + b − n 2 + c n + d . 6. n + a √ n + b − n + c √ n + d . 7. n 3 + a n2 + b − n 3 + c n2 + d . 8. n 2 + a n3 + b − n 2 + c n3 + d . 9. √ n + a √3 n + b − √ n + c √3 n + d . Exercise 1.1.9. Find the limits. 1. n 2 + a1n + a0 n + b − n 2 + c1n + c0 n + d . 2. n 2 + a1n + a0 n2 + b1n + b0 − n 2 + c1n + c0 n2 + d1n + d0 . 3. n + a n + b 2 − n + c n + d 2 . 4. n 2 + a n + b 2 − n 2 + c n + d 2 . Exercise 1.1.10. Find the limits, p, q > 0. 1. n p + a nq + b . 2. anp + bnq + c anq + bnp + c . 3. n p + a nq + b − n p + c nq + d . 4. n 2p + a1n p + a2 n2q + b1nq + b2 . 1.1.2 Sandwich Rule The following property reflects the intuition that if x and z are close to 3, then anything between x and z should also be close to 3. Proposition 1.1.4 (Sandwich Rule). Suppose xn ≤ yn ≤ zn for sufficiently big n. If limn→∞ xn = limn→∞ zn = l, then limn→∞ yn = l. Note that something holds for sufficiently big n is the same as something fails for only finitely many n. Example 1.1.3. By 2n − 3 > n for sufficiently big n (in fact, n > 3 is enough), we have 0 n 2 , and therefore 0 < √ n + 1 n − 1 < √ 2n n 2 = 2 √ 2 √ n