PLACEHOLDER Introduction to Limits

Learning Objectives

• Understand the intuitive concept of a limit
• Evaluate simple limits algebraically and graphically
• Recognize when limits do not exist
• Apply limit laws to solve problems

What is a Limit?

A limit describes the value that a function approaches as the input approaches a particular value.

We write: $$\lim_{x \to a} f(x) = L$$

This means: "As $x$ gets closer and closer to $a$, $f(x)$ gets closer and closer to $L$."

Graphical Understanding

Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$ when $x \neq 1$.

Even though $f(1)$ is undefined, we can ask: what value does $f(x)$ approach as $x$ approaches 1?

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1} (x+1) = 2$$

Limit Laws

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:

1. Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = L + M$

2. Difference Rule: $\lim_{x \to a} [f(x) - g(x)] = L - M$

3. Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$

4. Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)

5. Power Rule: $\lim_{x \to a} [f(x)]^n = L^n$

Types of Limits

Direct Substitution

When $f(x)$ is continuous at $x = a$:

$$\lim_{x \to a} f(x) = f(a)$$

Example: $\lim_{x \to 2} (3x^2 - 5x + 1) = 3(2)^2 - 5(2) + 1 = 3$

Indeterminate Forms

When direct substitution gives $\frac{0}{0}$, we need algebraic manipulation.

Example:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = \lim_{x \to 3} (x+3) = 6$$

One-Sided Limits

• Left-hand limit: $\lim_{x \to a^-} f(x)$ (approaching from the left)
• Right-hand limit: $\lim_{x \to a^+} f(x)$ (approaching from the right)

The limit $\lim_{x \to a} f(x)$ exists if and only if:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$

When Limits Don't Exist

1. Different one-sided limits
2. Infinite limits: $\lim_{x \to a} f(x) = \pm\infty$
3. Oscillating behavior: Function doesn't settle on a value

Important Limit

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

This limit is crucial for derivatives of trigonometric functions.

Practice Problems

Try these during class:

1. $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$
2. $\lim_{x \to 0} \frac{x^2 + 3x}{x}$
3. $\lim_{x \to 2^+} \frac{1}{x - 2}$