Understanding Limits and Continuity

📋 Course Outline

1. Limit Definition
2. Formal ε-δ Limit
3. Limit Evaluation Techniques
4. One-Sided Limits
5. Limits at Infinity
6. Continuity Conditions
7. Types of Discontinuities
8. Intermediate Value Theorem
9. Applications in Calculus
10. Limit and Continuity Summary

📖 1. Limit Definition

🔑 Key Concepts & Definitions

• Limit of a function: The value that (f(x)) approaches as (x) approaches a specific point (a). Denoted as (\lim_{x \to a} f(x)).
• Epsilon-Delta ((\varepsilon)-(\delta)) Definition: Formal criterion stating that for every (\varepsilon > 0), there exists a (\delta > 0) such that if (|x - a| < \delta), then (|f(x) - L| < \varepsilon). It rigorously defines what it means for a limit to exist.
• Finite Limit: When the limit approaches a specific real number (L) as (x) approaches (a).
• Infinite Limit: When (f(x)) grows without bound as (x) approaches (a), denoted as (\lim_{x \to a} f(x) = \infty).
• Limit at Infinity: The behavior of (f(x)) as (x \to \infty) or (-\infty), describing the end behavior of the function.