MATH 86857 Calculus of One Variable
Lecture 3 Notes
**Lecture 3: Limits and Continuity**
In this lecture, we delved deeper into the concepts of limits and continuity in calculus. These
topics are fundamental in understanding the behavior of functions and their properties.
**1. Limits:**
- The concept of limits is crucial in calculus as it helps us understand the behavior of functions
as they approach a certain point.
- Definition of a limit: A limit is the value that a function approaches as the input approaches a
particular value.
- We discussed left-hand and right-hand limits, where the function is approached from either
side of a point.
- Calculating limits algebraically involves techniques like direct substitution, factoring,
rationalizing, and conjugate multiplication.
- Special limits such as limits at infinity and infinite limits were also covered.
**2. Continuity:**
- A function is said to be continuous at a point if the limit of the function at that point exists and
is equal to the value of the function at that point.
- Types of discontinuities: We explored different types of discontinuities such as jump, infinite,
and removable discontinuities.
- Properties of continuous functions: Continuous functions exhibit properties such as the
intermediate value theorem, where the function takes on every value between two points.
**3. Theorems and Applications:**
- We covered the Squeeze Theorem, which helps in determining the limit of a function trapped
between two other functions.
- The Intermediate Value Theorem was discussed, stating that if a function is continuous on a
closed interval, it takes on every value between the function values at the endpoints.
- Applications of limits and continuity in real-world scenarios were highlighted, such as in
physics, engineering, and economics.
**4. Practice Problems:**
- To solidify our understanding, we worked through several practice problems involving limits
and continuity.
- Practicing these problems helps in honing our skills in evaluating limits and determining the
continuity of functions.
