Mastering Limit Concepts

📋 Course Outline

1. Definition of Limit
2. Graphical Representation
3. Limit Examples and Hints
4. Limit Calculation Methods
5. Key Limit Properties
6. Handling Special Cases
7. Organizing Limit Concepts

📖 1. Definition of Limit

🔑 Key Concepts & Definitions

• Limit of a function: The value that a function approaches as the input approaches a specific point.
  Mathematically: $\lim_{x \to a} f(x) = L$ means as $x$ gets closer to $a$, $f(x)$ gets closer to $L$.

• Approaching vs. reaching: Limits describe the behavior of a function near a point, not necessarily at the point itself.
  Example: $f(x) = \frac{\sin x}{x}$ as $x \to 0$, the limit exists even though $f(0)$ is undefined.

• One-sided limits: Limits considering only values approaching from the left ($x \to a^-$) or right ($x \to a^+$).
  Notation: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$.

• Infinite limits: When $f(x)$ increases or decreases without bound as $x$ approaches a point.
  Example: $\lim_{x \to 0} \frac{1}{x^2} = \infty$.

• Limit at infinity: Describes the behavior of $f(x)$ as $x \to \infty$ or $-\infty$.
  Example: $\lim_{x \to \infty} \frac{1}{x} = 0$.

📝 Essential Points