Fundamentals of Limits and Derivatives

📋 Course Outline

1. Limits
2. Derivative Definition
3. Differentiation Techniques
4. Differentiation Rules
5. Applications of Derivatives
6. Higher-Order Derivatives
7. Implicit Differentiation
8. Related Rates

📖 1. Limits

🔑 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) = L), meaning (f(x)) gets arbitrarily close to (L) when (x) is sufficiently close to (a).

• One-sided limits: The limit of (f(x)) as (x) approaches (a) from the left ((x \to a^-)) or from the right ((x \to a^+)). Both must agree for the two-sided limit to exist.

• Infinite limits: When (f(x)) increases or decreases without bound as (x) approaches (a), e.g., (\lim_{x \to a} f(x) = \infty) or (-\infty).

• Limit laws: Rules that allow the combination and simplification of limits, such as:

  • (\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x))
  • (\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x))

• Indeterminate forms: Expressions like (0/0) or (\infty/\infty) that require special techniques (e.g., algebraic manipulation, L'Hôpital's rule) to evaluate limits.

📝 Essential Points