Calculus 1, part 1 of 2: Limits and continuity

A total of 60 hours of lectures

This is an academic level course for university and college engineering. Due to its size, it is divided into two parts. This page describes the first of those two parts.

Level - Intermediate

You need to be familiar with the contents of the four Precalculus courses to comfortably follow along in this course.

Curriculum

Make sure that you check with your professor what parts of the course you will need for your 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.

Calculus 1, part 1 of 2

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Get the outline

A detailed list of all the lectures in part 1 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

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Course Objectives & Outcomes for part 1

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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.
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Arithmetic on the extended reals, and various types of indeterminate forms.
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Number sequences and their convergence or divergence; the epsilon-definition of limits of sequences, with illustrations and examples; accumulation points.
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Squeeze Theorem for sequences
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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.
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Limit of the sum, difference, product, quotient of two functions; limit of composition of two functions.
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Limits and continuity of elementary functions (polynomials, rational f., trigonometric and inverse trigonometric f., exponential, logarithmic and power f.).
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Some standard limits in the infinity: a comparison of polynomial growth (more generally: growth described by power f.), exponential, and logarithmic growth.
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Starting thinking about plotting functions: domain, range, behaviour around accumulation points outside the domain, asymptotes (vertical, horizontal, slant).
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The structure and properties of the set of real numbers as an ordered field with the Axiom of Completeness, and consequences of this definition.
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Supremum, infimum, and a reformulation of the Axiom of Completeness in these terms.
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Getting new limits from old limits: limit of the sum, difference, product, quotient, etc, of two sequences, with illustrations, formal proofs, and examples.
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Squeeze Theorem for functions
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Limits at infinity and infinite limits of functions: Cauchy’s definition (epsilon-delta) and Heine’s definition (sequential) of such limits; their equivalence.
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Properties of continuous functions: The Boundedness Theorem, The Max-Min Theorem, The Intermediate-Value Theorem.
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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.
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Continuous extensions and removable discontinuities; examples of discontinuous functions in one, several, or even infinitely many points in the domain.
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An introduction to more advanced topics: Cauchy sequences and their convergence; a word about complete spaces; limits and continuity in metric spaces.
Gustavo S.
Udemy student
excelent teacher!!!
Sandip K.
Udemy student
Excellent
Fernando L.
Udemy student
Super!
Murat F.
Udemy student
Another great course from Prof. Hania! Even though I took Calc 1 years ago, I still learned new topics and new approaches in limits.
Anonymized User
Udemy student
This Course is Awesome. She explains everything in such detail.
Mats E.
Udemy student
Very good teacher.
Vasanth G.
Udemy student
I am a beginner - higher mathematics is new to me. But this course is really good and I am really beginning to understand the concepts of real analysis. Very well structured course and repetitions of concepts in consequent videos is really making the basics imprinted in the mind. Thank you.
Dana N.
Udemy student
Love the detailed explanation!
Wanda W.
Udemy student
Very good course. Gives a lot of information about Real Analysis that are often omitted in other courses. It's really great to learn about the abstract axiomatic approach, clarified and illustrated with many practical examples. This is my 13th course by Hania and Martin that I've completed and I can recommend all of them.
Richard B.
Lecturer
The students from Hania's earlier courses, a prerequisite for the present, will not be disappointed, as she continues bringing life to what many might look upon as dried backpack provisions. Her enthusiasm for mathematical ideas and her care for passing these on are a living proof that the artefact of calculus – an exercise in accounting of a sort -- and the subtle art of mathematical analysis, may sometimes have something in common. The students not from Hania's earlier courses have not been forgotten either, as she amply revises the prerequisite material before getting technical. All in all, anybody serious enough about learning the rudiments of calculus and now looking for a thorough but humane introduction, needs only listen to Hania's presentations and honestly do the prescribed exercises.
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