Initial commit: Add Calculus Lesson 1 on Limits

README.md

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+# Calculus Lessons
+
+This repository contains a collection of calculus lessons in Markdown format with LaTeX equations.
+
+## Contents
+
+1. [Lesson 1: Introduction to Limits](calculus_lesson_1.md)
+
+## Viewing the Lessons
+
+These lessons use LaTeX math formatting which should render properly on Gitea when viewing the files.
+
+## Future Lessons
+
+- Continuity
+- Introduction to Derivatives
+- Rules of Differentiation
+- Applications of Derivatives
+- Introduction to Integration 

calculus_lesson_1.md

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+# Calculus I: Lesson 1 - Introduction to Limits
+
+## Objectives:
+- Understand the intuitive concept of a limit
+- Learn to evaluate limits graphically and numerically
+- Recognize when limits exist or don't exist
+- Solve basic limit problems algebraically
+
+## The Concept of Limits
+
+Welcome to Calculus I! Today we'll begin with limits, which form the foundation of calculus. A limit describes what a function approaches as the input approaches a certain value.
+
+The notation $\lim_{x \to a} f(x) = L$ means: "as x gets closer and closer to a (but not equal to a), f(x) gets closer and closer to L."
+
+## Exploring Limits Graphically
+
+Let's consider a function: $f(x) = \frac{x^2 - 4}{x - 2}$
+
+This function is undefined at x = 2 (division by zero). But what happens as x gets very close to 2?
+
+Let's explore by looking at values:
+
+| x approaches 2 from left | f(x) | x approaches 2 from right | f(x) |
+|--------------------------|------|---------------------------|------|
+| 1.9                      | 3.9  | 2.1                       | 4.1  |
+| 1.99                     | 3.99 | 2.01                      | 4.01 |
+| 1.999                    | 3.999| 2.001                     | 4.001|
+| 1.9999                   | 3.9999| 2.0001                   | 4.0001|
+
+As x gets closer to 2 (from either direction), f(x) gets closer to 4.
+
+We can simplify this function for x ≠ 2:
+$f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x+2$
+
+So, $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$
+
+**Important**: The limit exists even though f(2) is undefined. Limits concern the behavior near a point, not at the point itself.
+
+## One-Sided Limits
+
+Sometimes, a function approaches different values from the left and right.
+
+- Left-hand limit: $\lim_{x \to a^-} f(x)$ (approaching from values less than a)
+- Right-hand limit: $\lim_{x \to a^+} f(x)$ (approaching from values greater than a)
+
+For a limit to exist, both one-sided limits must exist and be equal:
+$\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$
+
+## Example: A Piecewise Function
+
+Consider:
+$$g(x) = 
+\begin{cases} 
+x^2, & \text{if } x < 1 \\
+3x-1, & \text{if } x \geq 1 
+\end{cases}$$
+
+Let's find $\lim_{x \to 1} g(x)$:
+
+- From the left: $\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} x^2 = 1$
+- From the right: $\lim_{x \to 1^+} g(x) = \lim_{x \to 1^+} (3x-1) = 3(1)-1 = 2$
+
+Since the left and right limits are different (1 ≠ 2), $\lim_{x \to 1} g(x)$ does not exist.
+
+## When Limits Don't Exist
+
+Limits don't exist when:
+1. Left and right limits are different (as in our example above)
+2. The function oscillates infinitely at the point (like $\sin(1/x)$ as x approaches 0)
+3. The function grows without bound (like $1/x^2$ as x approaches 0)
+
+## Basic Limit Laws
+
+If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:
+
+1. Sum: $\lim_{x \to a} [f(x) + g(x)] = L + M$
+2. Difference: $\lim_{x \to a} [f(x) - g(x)] = L - M$
+3. Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
+4. Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$
+5. Constant Multiple: $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$ for any constant c
+6. Power: $\lim_{x \to a} [f(x)]^n = L^n$ for integer n (assuming L > 0 if n is negative)
+
+## Common Techniques for Evaluating Limits
+
+1. **Direct Substitution**: If f is continuous at a, then $\lim_{x \to a} f(x) = f(a)$
+2. **Factoring**: Useful for addressing algebraic "holes" (like our first example)
+3. **Rationalization**: For limits involving square roots
+4. **Using known limits**: Like $\lim_{x \to 0} \frac{\sin x}{x} = 1$
+
+## Practice Problems
+
+Try these problems:
+
+1. $\lim_{x \to 3} (2x^2 - 5x + 1)$
+
+2. $\lim_{x \to 0} \frac{x^3 + 2x}{x}$
+
+3. $\lim_{x \to 2} \frac{x^2 - 3x + 2}{x-2}$
+
+4. $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$
+
+5. Find the values of x where this function is discontinuous:
+   $f(x) = \frac{x^2 - 9}{x+3}$
+
+## For Next Class
+
+- Read sections 1.1-1.3 in your textbook
+- Complete practice problems 1-15 in section 1.2
+- Start thinking about the concept of continuity, which we'll cover next time
+
+Remember: Limits form the foundation of calculus. Understanding them well will make derivatives and integrals much easier to grasp!