Calculus 1, part 1 of 2: Limits and continuity
Single variable calculus with elements of Real Analysis: from axioms and proofs to illustrations and computations
What you will 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.
Why take this course?
Calculus 1, part 1 of 2: Limits and continuity
Single variable calculus
S1. Introduction to the course
You will learn: about the content of this course, and generally about Calculus and its topics.
S2. Preliminaries: basic notions and elementary functions
You 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 formulas
You will learn: how to generalise some formulas with or without help of mathematical induction.
S4. The nature of the set of real numbers
You 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 limits
You 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 point
You 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 infinities
You 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 discontinuities
You will learn: continuous extensions and examples of removable discontinuity; piece-wise functions and their continuity or discontinuities.
S9. Properties of continuous functions
You 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 functions
You 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. Extras
You will learn: about all the courses we offer. 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.
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