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.
