Calculus 1, part 1 of 2: Limits and continuity

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).