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Peter Wittwer

Nous introduisons les fonctions réelles d'une variable réelle. Nous commençons par définir certaines de leurs propriétés, notamment la monotonie, la parité et la périodicité ainsi que les opérations entre fonctions. Nous définissons des fonctions particulières comme les fonctions hyperboliques. Nous continuons notre étude des fonctions, en définissant les fonctions définies par étapes, en particulier les fonctions Signum et Heaviside. Des manipulations importantes en pratique sur les fonctions sont les transformations affines. Nous rentrons finalement dans le cœur du sujet en définissant la limite épointée d'une fonction en un point et donnons des exemples de limites de fonctions. Nous finissons cette discussion par le concept de la limite à gauche et à droite. Dans la suite nous reprenons l'étude de la limite d’une fonctions en commençant par définir les opérations algébriques sur les limites. Nous étudions ensuite les limites infinies de fonctions. Afin de pouvoir calculer les limites de fonctions, nous donnons le théorème des deux gendarmes et nous discutons quelques exemples avec des fonctions algébriques, exponentielles et trigonométriques. Nous reprenons le concept de la limite épointée définie précédemment, en donnant une définition différente mais équivalente. Nous introduisons le concept de la continuité. Nous la définissons de deux manières différentes comme pour les limites de fonctions. Finalement, nous utilisons la continuité pour prolonger certaines fonctions, et nous étudions la continuité sur les intervalles ouverts.

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What's inside

Learning objectives

  • Terminologie, conventions
  • Les fonctions sinh(x) et cosh(x
  • Fonctions définies par étapes
  • Transformations affines
  • Définition de la limite (épointée)
  • Existence de la limite
  • Non existence de la limite
  • Limite à droite et à gauche
  • Opérations algébriques sur les limites
  • "limites infinies" et comportement à +/‐infinie
  • Théorème des deux gendarmes pour les fonctions
  • Définition de la limite avec epsilon et delta
  • Equivalence des définitions
  • Limite épointée et composition des fonctions
  • Définition de la continuité en un point
  • Fonctions continues et prolongement par conti nuité

Syllabus

Chapitre 5 : Limite d'une fonction
5.1 Terminologie, conventions
5.2 Définitions
5.3 Les fonctions sinh(x) et cosh(x
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5.4 Opérations algébriques
5.5 Exemples
5.6 Fonctions définies par étapes
5.7 Transformations affines
5.8 Motivation et définition de la limite épointée
5.9 Exemples
5.10 Limite à droite et à gauche
Chapitre 6 : Fonctions continues
6.1 Opérations algébriques sur les limites
6.2 "Limites infinies" et comportement à +/‐infinie
6.3 Théorème des deux gendarmes pour les fonctions
6.4 Exemples
6.5 Définition de la limite avec epsilon et delta
6.6 Démonstration (équivalence des définitions)
6.7 Limite épointée et composition des fonctions
6.8 Définition (continuité)
6.9 Définition de la continuité en un point par epsilon et delta
6.10 Fonctions continues et prolongement par continuité
6.11 Fonctions continues sur un intervalle ouvert

Good to know

Know what's good
, what to watch for
, and possible dealbreakers
Concepts et terminologie clés présentés au début du cours, ce qui facilite l'embarquement pour les nouveaux apprenants
Couvre les fonctions définies par étapes, utiles dans les applications pratiques
Définition rigoureuse de la limite d'une fonction, fondamentale pour l'analyse mathématique
Introduit les fonctions hyperboliques, utiles dans divers domaines
Traite du comportement des fonctions à l'infini, étendant la compréhension des limites
Englobe les transformations affines, essentielles pour la visualisation et la manipulation des fonctions

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Analyse I (partie 4) : Limite d'une fonction, fonctions continues with these activities:
Review your knowledge of functions
Reviewing the concept of functions, and their types will strengthen your familiarity and prepare you for the course materials
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  • Define a function
  • List the different types of functions
  • Graph a few functions
Review the concept of a function
Reviewing the basic concept of a function will help you better understand the more advanced concepts covered in this course.
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  • Define a function and its key components (domain, codomain, range, graph).
  • Identify different types of functions (e.g., linear, quadratic, exponential, logarithmic).
Watch tutorials on functions
Tutorials can help clarify topics often covered in this course and can help students with different learning styles process the information in new ways.
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  • Search for tutorials on functions
  • Watch several tutorials
  • Take notes on the key points
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Review Precalculus
Reviewing the concepts of Precalculus, such as functions, graphs, and trigonometry, will help you to build a stronger foundation for this course.
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  • Review the basics of functions, including domain and range.
  • Practice graphing functions, both linear and non-linear.
  • Review the basics of trigonometry, including trigonometric functions and identities.
Form a Study Group
Studying with a group can help you to stay motivated, share knowledge, and gain a deeper understanding of the material.
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  • Find a group of classmates who are also interested in studying together.
  • Set regular meeting times and stick to them.
  • Discuss the course material, work on problems together, and quiz each other.
Watch Video Lectures on Calculus
Watching video lectures can help you to learn at your own pace and reinforce the concepts that you are learning in class.
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  • Find video lectures on calculus from reputable sources.
  • Watch the lectures and take notes.
  • Review the lectures regularly to reinforce your understanding.
Practice identifying different types of functions
Practicing identifying different types of functions will help you develop a stronger understanding of their properties and behavior.
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  • Solve practice problems involving identifying different types of functions.
Solve limit problems
Practice solving limit problems to solidify your understanding of the concept.
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  • Review the definition and notation of a limit.
  • Practice finding limits of simple functions.
  • Use the limit laws to find limits of more complex functions.
  • Apply the limit definition to find limits of functions.
Simplify and evaluate functions
Practice simplifying and evaluating functions to strengthen your algebraic skills.
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  • Review the operations of algebra.
  • Simplify algebraic expressions involving functions.
  • Substitute values into functions.
Graph functions
Practice graphing functions to visualize their behavior and gain insights into their properties.
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  • Review the coordinate plane and graphing techniques.
  • Plot points and draw graphs by hand.
  • Use graphing tools to generate graphs of functions.
Explore online tutorials on limits and continuity
Gain a deeper understanding of limits and continuity by exploring online tutorials.
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  • Search for and identify reputable online tutorials.
  • Follow the tutorials carefully and take notes.
  • Complete the practice exercises provided in the tutorials.
Practice finding the limits of functions
Calculating limits of functions is a core concept that is required throughout the course. Practice will increase your speed and accuracy.
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  • Find the limit of a polynomial
  • Find the limit of a rational function
  • Find the limit of a logarithmic function
  • Find the limit of an exponential function
  • Find the limit of a trigonometric function
Join a study group
Working through problems with other students can help you see different perspectives and approaches.
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  • Find a study group
  • Meet regularly with the group
  • Work together on problems
Practice Solving Calculus Problems
Solving calculus problems regularly will help you to develop your problem-solving skills and gain a deeper understanding of the concepts.
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  • Find practice problems from textbooks or online resources.
  • Set aside time each week to practice solving problems.
  • Check your answers and identify areas where you need more practice.
Create a Concept Map
Creating a concept map will help you to visualize the relationships between different concepts and identify areas where you need more understanding.
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  • Identify the main concepts in the course.
  • Draw a diagram to represent the relationships between the concepts.
  • Add details and examples to help you understand the concepts.
Attend a Calculus Workshop
Attending a calculus workshop can provide you with an opportunity to learn from experts and get hands-on practice with the material.
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  • Find a calculus workshop that is offered by a reputable organization.
  • Register for the workshop and attend all of the sessions.
  • Participate in the activities and ask questions to enhance your understanding.
Write a summary of the key concepts
Solidify your understanding of the key concepts by writing a summary.
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  • Review the course materials.
  • Identify the key concepts.
  • Organize your thoughts and write a clear and concise summary.
Engage in discussions with peers
Clarify your understanding and learn from others by engaging in discussions with peers.
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  • Find a study group or discussion forum.
  • Participate in discussions and ask questions.
  • Share your insights and perspectives.
Tutor students
Reinforce your understanding and develop your teaching skills by tutoring students.
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  • Find opportunities to tutor students in the subject.
  • Prepare lesson plans.
  • Help students understand the concepts and solve problems.
Create a presentation on a specific function
By choosing a specific function to present on, you will deepen your knowledge of how functions work and better understand their special properties.
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  • Research your chosen function
  • Develop presentation materials
  • Practice your presentation
  • Deliver your presentation
Contribute to an open-source project
By contributing to open source projects, you will gain hands-on experience with functions and help improve the projects.
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  • Find an open-source project to contribute to
  • Install the project's development environment
  • Start making changes to the code
  • Submit your changes for review

Career center

Learners who complete Analyse I (partie 4) : Limite d'une fonction, fonctions continues will develop knowledge and skills that may be useful to these careers:
Engineering Manager
Engineering managers are responsible for the global operation and success of an engineering team. They direct team members, generate project schedules, track team progress, and execute the project. This course, Analyze I Part 4: Limits of a Function, Continuous Functions, will help you develop the mathematical and problem-solving skills that are important in engineering management. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Electrical Engineer
Electrical engineers design, develop, test, and supervise the installation of electrical systems. They may work in a variety of industries, including power generation, transmission, and distribution; manufacturing; and telecommunications. This course will help you develop the mathematical and problem-solving skills that are essential for success in electrical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Civil Engineer
Civil engineers design, build, and maintain the physical infrastructure of our world, including roads, bridges, buildings, and water systems. They may work in a variety of industries, including construction, transportation, and environmental protection. This course will help you develop the mathematical and problem-solving skills that are essential for success in civil engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Materials Engineer
Materials engineers research, develop, and test new materials for use in a variety of applications, such as aerospace, automotive, and medical devices. They may work in a variety of industries, including manufacturing, research, and development. This course will help you develop the mathematical and problem-solving skills that are essential for success in materials engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Industrial Engineer
Industrial engineers design, improve, and install integrated systems of people, materials, information, equipment, and energy. They may work in a variety of industries, including manufacturing, healthcare, and logistics. This course will help you develop the mathematical and problem-solving skills that are essential for success in industrial engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Mechanical Engineer
Mechanical engineers design, develop, and test mechanical systems, such as engines, machines, and robots. They may work in a variety of industries, including manufacturing, transportation, and energy. This course will help you develop the mathematical and problem-solving skills that are essential for success in mechanical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Biomedical Engineer
Biomedical engineers design, develop, and test medical devices and systems. They may work in a variety of industries, including healthcare, pharmaceuticals, and medical research. This course will help you develop the mathematical and problem-solving skills that are essential for success in biomedical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Chemical Engineer
Chemical engineers design, develop, and operate chemical plants and processes. They may work in a variety of industries, including chemicals, pharmaceuticals, and food processing. This course will help you develop the mathematical and problem-solving skills that are essential for success in chemical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Aerospace Engineer
Aerospace engineers design, develop, test, and operate aircraft, spacecraft, and other vehicles that travel through the air. They may work in a variety of industries, including aviation, defense, and space exploration. This course will help you develop the mathematical and problem-solving skills that are essential for success in aerospace engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Computer Engineer
Computer engineers design, develop, and test computer systems and applications. They may work in a variety of industries, including software development, hardware manufacturing, and IT consulting. This course will help you develop the mathematical and problem-solving skills that are essential for success in computer engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
Economist
Economists study the production, distribution, and consumption of goods and services. They may work for a variety of organizations, including government agencies, think tanks, and universities. This course will help you develop the mathematical and problem-solving skills that are essential for success as an economist. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world economic problems.
Actuary
Actuaries use mathematics to assess risk and uncertainty. They may work for a variety of organizations, including insurance companies, pension funds, and government agencies. This course will help you develop the mathematical and problem-solving skills that are essential for success as an actuary. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world actuarial problems.
Statistician
Statisticians collect, analyze, and interpret data. They may work for a variety of organizations, including government agencies, businesses, and non-profit organizations. This course will help you develop the mathematical and problem-solving skills that are essential for success as a statistician. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world statistical problems.
Data Scientist
Data scientists use data to solve business problems. They may work for a variety of organizations, including technology companies, financial institutions, and healthcare providers. This course will help you develop the mathematical and problem-solving skills that are essential for success as a data scientist. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world data science problems.
Financial Analyst
Financial analysts research and analyze financial data to make investment recommendations. They may work for a variety of organizations, including investment banks, hedge funds, and asset management companies. This course will help you develop the mathematical and problem-solving skills that are essential for success as a financial analyst. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world financial problems.

Reading list

We've selected 12 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Analyse I (partie 4) : Limite d'une fonction, fonctions continues.
Cet ouvrage complet offre une couverture approfondie des limites et de la continuité. Il est particulièrement précieux pour les étudiants avancés et les chercheurs qui cherchent à approfondir leurs connaissances.
Ce livre classique est une référence incontournable pour l'analyse mathématique. Il couvre les limites, la continuité et d'autres concepts fondamentaux, avec une approche rigoureuse et approfondie.
Ce livre classique fournit une base solide pour l'analyse mathématique, y compris les limites et la continuité. Il est particulièrement utile pour acquérir une compréhension approfondie des concepts fondamentaux.
Ce livre classique est une ressource précieuse pour les étudiants avancés et les professionnels. Il fournit une exposition rigoureuse des limites, de la continuité et d'autres concepts fondamentaux.
Ce livre est conçu pour les étudiants diplômés. Il fournit une présentation rigoureuse et approfondie des limites, de la continuité et d'autres sujets d'analyse réelle.
Cet ouvrage accessible fournit une introduction complète à l'analyse réelle, couvrant les limites, la continuité et d'autres concepts fondamentaux. Il est utile pour approfondir la compréhension et fournir des exemples supplémentaires.
Ce livre fournit une introduction complète à l'analyse réelle, y compris les limites et la continuité. Il est particulièrement utile pour les étudiants qui se préparent aux études supérieures.
Ce livre propose une approche rigoureuse et approfondie du calcul, y compris les limites et la continuité. Il est précieux pour développer une compréhension conceptuelle solide et acquérir des compétences techniques.
Ce recueil d'exercices propose une variété de problèmes sur les limites, la continuité et d'autres sujets d'analyse mathématique. Il est utile pour consolider la compréhension et développer les compétences techniques.
Ce texte concis et clair fournit une introduction complète à l'analyse mathématique, couvrant les limites, la continuité et d'autres sujets fondamentaux. Il est utile en tant que référence rapide et pour consolider la compréhension.
Ce manuel fournit une introduction claire et accessible à l'analyse élémentaire, couvrant les limites et la continuité. Il est utile pour les étudiants débutants et ceux qui cherchent à renforcer leur compréhension fondamentale.

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