From 7a940ead3c6e44a274f08878a0631aa4def8d309 Mon Sep 17 00:00:00 2001 From: Hugh Ratsch Date: Sun, 23 Mar 2025 20:09:52 -0500 Subject: [PATCH] Initial commit: Add Calculus Lesson 1 on Limits --- README.md | 19 ++++++++ calculus_lesson_1.md | 111 +++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 130 insertions(+) create mode 100644 README.md create mode 100644 calculus_lesson_1.md diff --git a/README.md b/README.md new file mode 100644 index 0000000..fe379a4 --- /dev/null +++ b/README.md @@ -0,0 +1,19 @@ +# 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 \ No newline at end of file diff --git a/calculus_lesson_1.md b/calculus_lesson_1.md new file mode 100644 index 0000000..4d11f6b --- /dev/null +++ b/calculus_lesson_1.md @@ -0,0 +1,111 @@ +# 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! \ No newline at end of file