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Lecture 2: Limits and Continuity, Study notes of Calculus

Recap of the previous lecture on the concept of functions in calculus. Introduction to the fundamental concepts of limits and continuity in calculus

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2023/2024

Uploaded on 04/15/2025

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MATH 86857 Calculus of One Variable
Lecture 2 Notes
**MATH 86857 Calculus of One Variable**
**Lecture 2: Limits and Continuity**
**Introduction:**
- Recap of the previous lecture on the concept of functions in calculus.
- Introduction to the fundamental concepts of limits and continuity in calculus.
**Limits:**
- Definition of a limit: The limit of a function as it approaches a certain point.
- Understanding the notation of limits: lim x->a f(x) = L.
- Exploring the concept of one-sided limits and their significance.
- Evaluating limits algebraically and graphically.
- Properties of limits: sum, product, quotient, and composition of functions.
- Solving limit problems using direct substitution and factoring techniques.
**Continuity:**
- Definition of continuity: A function is continuous at a point if the limit of the function at that
point exists and is equal to the function's value at that point.
- Understanding the three conditions for continuity: a function is continuous at a point, over an
interval, and on a domain.
- Types of discontinuities: removable, jump, and infinite discontinuities.
- Identifying and analyzing continuity using graphs and algebraic techniques.
- The Intermediate Value Theorem and its application in determining the existence of roots of
functions.
**Practical Applications:**
- Real-world examples of limits and continuity in various fields such as physics, engineering,
and economics.
- Importance of understanding limits and continuity in calculus for solving practical problems.
**Conclusion:**
- Recap of key concepts covered in the lecture: limits, continuity, types of discontinuities, and
practical applications.
- Preview of the next lecture on derivatives and their applications.
**Additional Resources:**
- Recommended textbook readings on limits and continuity.
- Practice problems and exercises to reinforce understanding.
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MATH 86857 Calculus of One Variable

Lecture 2 Notes

MATH 86857 Calculus of One Variable Lecture 2: Limits and Continuity Introduction:

  • Recap of the previous lecture on the concept of functions in calculus.
  • Introduction to the fundamental concepts of limits and continuity in calculus. Limits:
  • Definition of a limit: The limit of a function as it approaches a certain point.
  • Understanding the notation of limits: lim x->a f(x) = L.
  • Exploring the concept of one-sided limits and their significance.
  • Evaluating limits algebraically and graphically.
  • Properties of limits: sum, product, quotient, and composition of functions.
  • Solving limit problems using direct substitution and factoring techniques. Continuity:
  • Definition of continuity: A function is continuous at a point if the limit of the function at that point exists and is equal to the function's value at that point.
  • Understanding the three conditions for continuity: a function is continuous at a point, over an interval, and on a domain.
  • Types of discontinuities: removable, jump, and infinite discontinuities.
  • Identifying and analyzing continuity using graphs and algebraic techniques.
  • The Intermediate Value Theorem and its application in determining the existence of roots of functions. Practical Applications:
  • Real-world examples of limits and continuity in various fields such as physics, engineering, and economics.
  • Importance of understanding limits and continuity in calculus for solving practical problems. Conclusion:
  • Recap of key concepts covered in the lecture: limits, continuity, types of discontinuities, and practical applications.
  • Preview of the next lecture on derivatives and their applications. Additional Resources:
  • Recommended textbook readings on limits and continuity.
  • Practice problems and exercises to reinforce understanding.
  • Office hours and tutoring services for additional assistance. Note:
  • It is essential to practice solving problems related to limits and continuity to master these fundamental concepts in calculus.