calculus-lessons/calculus_lesson_1.md

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 the function g(x) defined as:

g(x) = x^2 \text{ for } x < 1 g(x) = 3x-1 \text{ for } x \geq 1

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 \frac{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!