Calculus 1, Part 1 Of 2: Limits And Continuity

Single variable calculus

What you'll learn

How to solve problems concerning limits and continuity of real-valued functions of 1 variable (illustrated with 491 solved problems) and why these methods work.

The structure and properties of the set of real numbers as an ordered field with the Axiom of Completeness, and consequences of this definition.

Arithmetic on the extended reals, and various types of indeterminate forms.

Supremum, infimum, and a reformulation of the Axiom of Completeness in these terms.

Number sequences and their convergence or divergence; the epsilon-definition of limits of sequences, with illustrations and examples; accumulation points.

Getting new limits from old limits: limit of the sum, difference, product, quotient, etc, of two sequences, with illustrations, formal proofs, and examples.

Squeeze Theorem for sequences

Squeeze Theorem for functions

The concept of a finite limit of a real-valued function of one real variable in a point: Cauchy's definition, Heine's definition; proof of their equivalence.

Limits at infinity and infinite limits of functions: Cauchy's definition (epsilon-delta) and Heine's definition (sequential) of such limits; their equivalence.

Limit of the sum, difference, product, quotient of two functions; limit of composition of two functions.

Properties of continuous functions: The Boundedness Theorem, The Max-Min Theorem, The Intermediate-Value Theorem.

Limits and continuity of elementary functions (polynomials, rational f., trigonometric and inverse trigonometric f., exponential, logarithmic and power f.).

Some standard limits in zero: sin(x)/x, tan(x)/x, (e^x-1)/x, ln(x+1)/x and a glimpse into their future applications in Differential Calculus.

Some standard limits in the infinity: a comparison of polynomial growth (more generally: growth described by power f.), exponential, and logarithmic growth.

Continuous extensions and removable discontinuities; examples of discontinuous functions in one, several, or even infinitely many points in the domain.

Starting thinking about plotting functions: domain, range, behaviour around accumulation points outside the domain, asymptotes (vertical, horizontal, slant).

An introduction to more advanced topics: Cauchy sequences and their convergence; a word about complete spaces; limits and continuity in metric spaces.

Requirements

"Precalculus 1: Basic notions" (or equivalent): mathematical notation, logic, sets, proofs

"Precalculus 2: Polynomials and rational functions" (or equivalent)

"Precalculus 3: Trigonometry" (or equivalent)

"Precalculus 4: Exponentials and logarithms" (or equivalent)

You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

Description

Calculus 1, part 1 of 2: Limits and continuitySingle variable calculusS1. Introduction to the courseYou will learn: about the content of this course, and generally about Calculus and its topics.S2. Preliminaries: basic notions and elementary functionsYou will learn: you will get a brief recap of the Precalculus stuff you are supposed to master in order to be able to follow Calculus, but you will also get some words of consolation and encouragement, I promise.S3. Some reflections about the generalising of formulasYou will learn: how to generalise some formulas with or without help of mathematical induction.S4. The nature of the set of real numbersYou will learn: about the structure and properties of the set of real numbers as an ordered field with the Axiom of Completeness, and consequences of this definition.S5. Sequences and their limitsYou will learn: the concept of a number sequence, with many examples and illustrations; subsequences, monotone sequences, bounded sequences; the definition of a limit (both proper and improper) of a number sequence, with many examples and illustrations; arithmetic operations on sequences and The Limit Laws for Sequences; accumulation points of sequences; the concept of continuity of arithmetic operations, and how The Limit Laws for Sequences will serve later in Calculus for computing limits of functions and for proving continuity of elementary functions; Squeeze Theorem for Sequences; Weierstrass' Theorem about convergence of monotone and bounded sequences; extended reals and their arithmetic; determinate and indeterminate forms and their importance; some first insights into comparing infinities (Standard Limits in the Infinity); a word about limits of sequences in metric spaces; Cauchy sequences (fundamental sequences) and a sketch of the construction of the set of real numbers using an equivalence relation on the set of all Cauchy sequences with rational elements.S6. Limit of a function in a pointYou will learn:  the concept of a finite limit of a real-valued function of one real variable in a point: Cauchy's definition, Heine's definition (aka Sequential condition), and their equivalence; accumulation points (limit points, cluster points) of the domain of a function; one-sided limits; the concept of continuity of a function in a point, and continuity on a set; limits and continuity of elementary functions as building blocks for all the other functions you will meet in your Calculus classes; computational rules: limit of sum, difference, product, quotient of two functions; limit of a composition of two functions; limit of inverse functions; Squeeze Theorem; Standard limits in zero and other methods for handling indeterminate forms of the type 0/0 (factoring and cancelling, using conjugates, substitution).S7. Infinite limits and limits in the infinitiesYou will learn: define and compute infinite limits and limits in infinities for functions, and how these concepts relate to vertical and horizontal asymptotes for functions; as we already have learned the arithmetic on extended reals in Section 5, we don't need much theory here; we will perform a thorough analysis of limits of indeterminate forms involving rational functions in both zero and the infinities.S8. Continuity and discontinuitiesYou will learn: continuous extensions and examples of removable discontinuity; piece-wise functions and their continuity or discontinuities.S9. Properties of continuous functionsYou will learn: basic properties of continuous functions: The Boundedness Theorem, The Max-Min Theorem, The Intermediate-Value Theorem; you will learn the formulation and the meaning of these theorems, together with their proofs (in both written text and illustrations) and examples of their applications; we will revisit some old examples from the Precalculus series where we used these properties without really knowing them in a formal way (but well relying on our intuition, which is not that bad at a Precalculus level); uniform continuity; a characterisation of continuity with help of open sets.S10. Starting graphing functionsYou will learn: how to start the process of graphing real-valued functions of one real variable: determining the domain and its accumulation points, determining the behaviour of the function around the accumulation points of the domain that are not included in the domain, determining points of discontinuity and one-sided limits in them, determining asymptotes. We will continue working with this subject in "Calculus 1, part 2 of 2: Derivatives with applications".S11. ExtrasYou will learn: about all the courses we offer, and where to find discount coupons. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 225 videos and their titles, and with the texts of all the 491 problems solved during this course, is presented in the resource file “001 List_of_all_Videos_and_Problems_Calculus_1_p1.pdf” under video 1 ("Introduction to the course"). This content is also presented in video 1.

Overview

Section 1: Introduction to the course

Lecture 1 Introduction to the course

Lecture 2 What is Calculus and who needs it

Lecture 3 The difference between Calculus and Real Analysis; my choices

Lecture 4 The greatest names in Calculus

Lecture 5 Elementary functions and their superpowers

Lecture 6 How we cheated our way through the high school while discussing functions

Lecture 7 What we are going to learn in this series Calculus 1

Lecture 8 Limits are at the heart and soul of Calculus

Section 2: Preliminaries: Basic notions and elementary functions

Lecture 9 How essential is it to master Precalculus before Calculus?

Lecture 10 The essence of Precalculus 1

Lecture 11 The essence of Precalculus 2

Lecture 12 The essence of Precalculus 3

Lecture 13 The essence of Precalculus 4

Lecture 14 Ask questions, use QA

Section 3: Some reflections about the generalising of formulas

Lecture 15 Introduction to Section 3

Lecture 16 Induction: a brief repetition

Lecture 17 Generalisations of 3 basic laws (associativity, commutativity, distributivity)

Lecture 18 Some examples, before we get to the general method; Ex1

Lecture 19 Some examples, before we get to the general method; Ex2

Lecture 20 Some examples, before we get to the general method; Ex3

Lecture 21 Types of formulas which expand easily because of the associativity of operations

Lecture 22 An exercise: applying The General Method

Lecture 23 Optional: Linear transformations between vector spaces

Lecture 24 Optional, Future: Limits, differentiation, and other linear operations

Lecture 25 A word about the sigma symbol and The Binomial Theorem

Lecture 26 Squaring the sums

Lecture 27 Less obvious, but possible: when the LHS is a function of a sum or of a product

Lecture 28 Future: The derivative of a product formula

Lecture 29 Two trigonometric formulas to remember; a complex-numbers trick

Lecture 30 Generalisations of the formulas to remember for the sum of more arguments

Lecture 31 How to get more trigonometric formulas for free

Lecture 32 Derivation versus proof by induction; Formula 1

Lecture 33 Derivation versus proof by induction; Formula 2

Lecture 34 Future: Riemann integral and area under the graph

Lecture 35 Derivation versus proof by induction; Formula 3

Lecture 36 Extremely important: a generalisation of the triangle inequality

Lecture 37 Bernoulli's inequality

Lecture 38 An interesting lemma for the future (for the proof of Stolz-Cesàro Theorem)

Section 4: The nature of the set of real numbers

Lecture 39 How to use this section; good news first!

Lecture 40 The theory of real numbers justifies the practical consequences

Lecture 41 Where in the Precalculus series you find information about real numbers

Lecture 42 Algebra, orders, and completeness

Lecture 43 About deriving rules directly from the axioms

Lecture 44 Optional: Deriving rules directly from the axioms; addition

Lecture 45 Optional: Deriving rules directly from the axioms; multiplication

Lecture 46 Optional: Deriving rules directly from the axioms; multiplication (continued)

Lecture 47 Optional: Deriving rules directly from the axioms; inequalities

Lecture 48 Optional: Some examples of fields: R, Q, C, Zp

Lecture 49 Just for fun: There are plenty of ordered number fields between Q and R

Lecture 50 Absolute value, distances, and Triangle Inequality

Lecture 51 Some definitions with examples: supremum, infimum, maximum, minimum

Lecture 52 Axiom of Completeness and its reformulation in terms of supremums

Lecture 53 Hardcore: Existence of roots

Lecture 54 A word about Peano axioms defining the set of natural numbers

Lecture 55 The Minimum Principle for natural numbers

Lecture 56 A word about construction of integer, rational, and real numbers

Lecture 57 The floor function, or the greatest integer function

Lecture 58 Three equivalent properties following from the Axiom of Completeness

Lecture 59 Density of Q in R, and why we need to know about it

Lecture 60 Hardcore: There is just one set of real numbers

Lecture 61 Supremum, infimum, etc; Example 1

Lecture 62 Supremum, infimum, etc; Example 2

Lecture 63 A preparation for some subtleties in the definitions of limits and continuity

Lecture 64 The magical power of leading themes

Lecture 65 Accumulation points (cluster points) and isolated points; derived sets

Lecture 66 Your first encounter with our leading theme

Lecture 67 Various relations between the concepts of supremum, infimum, accumulation point

Section 5: Sequences and their limits

Lecture 68 Sequences in Precalculus 1

Lecture 69 More reasons (than given in V8) to study sequences now

Lecture 70 What is a sequence? Notation and terminology

Lecture 71 Sequences as functions; various ways of defining sequences

Lecture 72 Exercise 1: Reading formulas (The one with MANIM for multiple purposes)

Lecture 73 Exercise 2: Finding formulas

Lecture 74 Exercise 3: Guess and prove

Lecture 75 Exercise 4: Guess and prove

Lecture 76 Problem 1: Simplify the formula

Lecture 77 Optional: Problem 2: Recursive to explicit (or: closed) formula

Lecture 78 Number sequences versus functions defined for all positive arguments

Lecture 79 Bounded sequences

Lecture 80 Monotone sequences versus monotone functions

Lecture 81 Arithmetic operations on sequences

Lecture 82 Two ways of depicting sequences; prelude to convergence

Lecture 83 Playing with symbols

Lecture 84 Accumulation points of sequences

Lecture 85 Accumulation points of sequences, Exercise 5

Lecture 86 What is a subsequence? Some examples

Lecture 87 Limit of a sequence, definition and notation

Lecture 88 It doesn't matter what happens with the first m elements

Lecture 89 Accumulation points are limits of subsequences

Lecture 90 If a sequence is convergent then it has exactly one accumulation point

Lecture 91 Limit of a sequence; playing with epsilons

Lecture 92 Computing limits from the definition, Exercise 6

Lecture 93 Computing limits from the definition, Exercise 7

Lecture 94 Computing limits from the definition, Exercise 8

Lecture 95 Why you need proofs with epsilons; some preparations for them

Lecture 96 Properties of convergent sequences

Lecture 97 Squeeze Theorem for sequences

Lecture 98 Squeeze Theorem for sequences, Exercise 9

Lecture 99 Squeeze Theorem for sequences, Exercise 10

Lecture 100 Extremely important: New limits from old limits

Lecture 101 New limits from old limits, Exercise 11

Lecture 102 New limits from old limits, Exercise 12

Lecture 103 New limits from old limits, Exercise 13

Lecture 104 New limits from old limits, proof part 1

Lecture 105 New limits from old limits, proof part 2

Lecture 106 New limits from old limits, proof part 3

Lecture 107 New limits from old limits, proof part 4

Lecture 108 Weierstrass' Theorem about convergence of bounded monotone sequences

Lecture 109 Weierstrass' Theorem, Example 0

Lecture 110 Weierstrass' Theorem, Exercise 14

Lecture 111 Monotone, bounded, convergent sequences; a test

Lecture 112 Extended reals

Lecture 113 What does it mean that arithmetic operations are continuous?

Lecture 114 Improper limits

Lecture 115 Extending arithmetic to extended reals

Lecture 116 Powers involving extended reals

Lecture 117 Some important examples, Exercise 15

Lecture 118 Very important: Indeterminate forms

Lecture 119 Comparing infinities: Not always intuitive; Exercise 16

Lecture 120 Comparing infinities: An important limit for the exponential function

Lecture 121 Comparing infinities: More quotients

Lecture 122 Optional: Cauchy sequences, sequences in metric spaces, and completeness

Section 6: Limit of a function in a point

Lecture 123 It's all about proximity, also for functions

Lecture 124 In what kind of points is it meaningful to examine limits of a function?

Lecture 125 An example from Precalculus 1

Lecture 126 An example from Precalculus 2

Lecture 127 An example from Precalculus 3

Lecture 128 An example from Precalculus 4

Lecture 129 Formal definition of a limit in a point, with an illustration

Lecture 130 Limit, if exists, is unique

Lecture 131 Limits of some functions, Example 1

Lecture 132 Limits of some functions, Example 2

Lecture 133 Limits of some functions, Example 3

Lecture 134 Some (earlier) examples where the limit does not exist

Lecture 135 One-sided limits

Lecture 136 One-sided limits, examples

Lecture 137 One-sided limits, more examples

Lecture 138 Extremely important: New limits from old limits

Lecture 139 New limits from old limits, an exercise

Lecture 140 Another approach to the topic of limits

Lecture 141 Heine's definition of limits, and a remark about notation

Lecture 142 Equivalence of Cauchy's and Heine's definitions of limits

Lecture 143 Two proofs of the powerful theorem from V138

Lecture 144 What does it mean that a function is continuous?

Lecture 145 A handy test for continuity

Lecture 146 Examining continuity, some examples

Lecture 147 Some warnings about different definitions of continuity and discontinuity

Lecture 148 Continuity of polynomials, rational functions, and power functions

Lecture 149 Continuity of trigonometric functions

Lecture 150 An important lemma about one-sided limits of monotone functions

Lecture 151 Continuity of exponential functions

Lecture 152 Compositions of continuous functions

Lecture 153 Compositions of continuous functions: some examples

Lecture 154 Continuity of inverse functions

Lecture 155 Continuity of inverse functions: why the assumption about interval is important

Lecture 156 Some important consequences of the theorems in V152 and 154

Lecture 157 From determinate to indeterminate forms: something needs to be done

Lecture 158 Squeeze Theorem for functions

Lecture 159 An application of Squeeze Theorem: a standard limit in zero

Lecture 160 Now we can finally motivate the formula for the area of a disk

Lecture 161 More standard limits in zero

Lecture 162 Problem solving, Problem 1

Lecture 163 Problem solving, Problem 2

Lecture 164 Problem solving, Problem 3

Lecture 165 Problem solving, Problem 4

Lecture 166 Problem solving, Problem 5

Lecture 167 Problem solving, Problem 6

Lecture 168 Problem solving, Problem 7

Lecture 169 Problem solving, Problem 8

Section 7: Infinite limits and limits in the infinities

Lecture 170 An example showing (almost) all the concepts from this section

Lecture 171 We do have all the theory needed

Lecture 172 What about all the warnings?

Lecture 173 Infinite limits in the infinities

Lecture 174 Infinite limits in the infinities, a sequential condition

Lecture 175 Finite limits in the infinities

Lecture 176 Finite limits in the infinities, a sequential condition

Lecture 177 Horizontal asymptotes

Lecture 178 Infinite limits at accumulation points of the domain (outside the domain)

Lecture 179 Infinite limits at accumulation points of the domain, a sequential condition

Lecture 180 Vertical asymptotes

Lecture 181 Tangent and arctangent, and their asymptotes

Lecture 182 Comparing infinities: as in V119

Lecture 183 Different horizontal asymptotes in plus and minus infinity

Lecture 184 Limits, Problem 1

Lecture 185 Limits, Problem 2

Lecture 186 Limits, Problem 3

Lecture 187 Limits, Problem 4

Lecture 188 Limits, Problem 5

Lecture 189 Limits, Problem 6

Section 8: Continuity and discontinuities

Lecture 190 Three types of discontinuities, continuous extensions, and some warnings

Lecture 191 Continuity and discontinuities, Problem 1

Lecture 192 Continuity and discontinuities, Problem 2

Lecture 193 Continuity and discontinuities, Problem 3

Lecture 194 Continuity and discontinuities, Problem 4

Lecture 195 Continuity and discontinuities, Problem 5

Lecture 196 Continuity and discontinuities, Problem 6

Lecture 197 Continuity and discontinuities, Problem 7

Lecture 198 Continuity and discontinuities, Problem 8

Lecture 199 Optional: Examples of functions with various numbers of discontinuity points

Lecture 200 Piece-wise functions where the ends will always meet

Section 9: Properties of continuous functions

Lecture 201 Briefly about important properties of functions continuous on intervals

Lecture 202 Separation lemma

Lecture 203 The Boundedness Theorem

Lecture 204 The Max-Min Theorem

Lecture 205 Halving intervals

Lecture 206 The Intermediate-Value Theorem

Lecture 207 Some examples from Precalculus

Lecture 208 Properties of continuous functions, Problem 1

Lecture 209 Properties of continuous functions, Problem 2

Lecture 210 Properties of continuous functions, Problem 3

Lecture 211 Properties of continuous functions, Problem 4

Lecture 212 Properties of continuous functions, Problem 5

Lecture 213 Properties of continuous functions, Problem 6

Lecture 214 Uniform continuity

Lecture 215 Future: Open, closed, compact, and connected sets in metric spaces

Lecture 216 Future: Reformulation of the three important theorems in new terms

Lecture 217 Advanced: Three characterisations of continuous functions

Section 10: Starting graphing functions

Lecture 218 A todo list for plotting functions: for now and for the future

Lecture 219 Slant asymptotes, and a clarity for rational functions

Lecture 220 A rational function, Problem 1

Lecture 221 Different horizontal asymptotes in the infinities, Problem 2

Lecture 222 Asymptotes, Problem 3

Lecture 223 Asymptotes, Problem 4

Lecture 224 One-sided vertical asymptote, Problem 5

Lecture 225 Wrap-up Calculus 1, part 1 of 2

Section 11: Extras

Lecture 226 Bonus Lecture

University and college students wanting to learn Single Variable Calculus (or Real Analysis),High school students curious about university mathematics; the course is intended for purchase by adults for these students