Basic Calculus

What is the limit of the function $rac{x^2 - 9}{x - 3}$ as $x$ approaches 3?

Using substitution, what is the value of the limit $rac{x + 2}{x o 4}$?

Which of the following statements about the limit of $rac{x^2 - 4}{x - 2}$ as $x$ approaches 2 is correct?

What is the correct approach to finding the limit of the function $f(x) = 2x + 1$ as $x$ approaches 2?

If the limit of $rac{x^2 - 9}{x - 3}$ is indeterminate, what must be done to resolve it?

What does the limit of a function represent as the variable approaches a particular value?

In the expression lim f(x) = L as x approaches c, what does L represent?

What can be concluded if for lim (3x - 2) as x approaches 2, the estimated values are 3.7, 3.97, 3.997, ?, 4.003, 4.03?

If you substitute x = 3 in the limit lim (2/(3-x)) as x approaches 3, what is the result?

Which statement is true about the behavior of functions as x approaches a constant c?

Basic Calculus - Limits

Limits are fundamental to calculus, considered the backbone of the field, often referred to as the Mathematics of Change.
Studying limits is crucial for understanding change in detail.
Evaluating a limit forms the basis for calculating derivatives and integrals.

Defining Limits

Limits focus on the behavior of a function as its variable approaches a specific value (a constant).
The variable's value can only get very close to the constant but never equal the constant.
The limit clarifies the function's behavior near the constant.

Definition of Limit

If a function, f(x), gets arbitrarily close to a unique number L as x approaches c from either direction, then the limit of f(x) as x approaches c is L.

Notation

The limit of f(x) as x approaches c is written as lim f(x) = L x→c

Examples of Limits

Calculating limits numerically (using values close to c)
Calculating limits algebraically (substitution and factoring)
- Example: Limit of (3x² as x approaches 4 = 48
- Example: Limit of 2x / (x² + 1) as x approaches 0 = 0
Example of a limit that does not exist involves division by zero

Classroom Rules

Raise your hand to answer questions.
Do not use cell phones during class.
Respect your fellow classmates.

Examples of Calculating Limits

Example 1: numerically estimating lim(3x – 2) as x approaches 2 = 4
Example 2: substitution method to find that lim(2/(3 – x) as x approaches 3 does not exist (undefined).
Example 3: factoring and substitution to show lim ((x² – 9)/(x – 3)) as x approaches 3 = 6
Example 4: numerically and algebraically determining lim(x + 2) as x approaches 4 = 6

Additional Problems

Problem 1: Find the limit of (2x + 1) as x approaches 2 via substitution = 5
Problem 2: Find the limit of (2x – 6) as x approaches 4 via substitution = 2
Problem 3: Find the limit of ((x² – 4)/(x – 2)) as x approaches 2 (use factoring) = 4

Explore the fundamental concept of limits in calculus, which serves as the foundation for understanding derivatives and integrals. This quiz will help you grasp the behavior of functions as variables approach specific values through various methods of evaluation.

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