calculus 2-1

This document captures the key concepts and explanations provided in the transcript regarding factoring, limits, and continuity, which are fundamental topics in algebra and calculus.

Definition of Factoring: To factor means to break an expression into two sections such that each section is linear.
Step 1: Check the Leading Coefficient
- Example: The leading coefficient is 1 in the polynomial expression being discussed.
Finding Numbers to Factor
- To factor a trinomial, find two numbers that:
- Add to give the coefficient of the linear term (middle term).
- Multiply to give the constant term (last term).
- Example: For x² + 7x + 12,
- Find two numbers that add up to 7 and multiply to 12: These numbers are 3 and 4.
- Factored form: (x + 3)(x + 4).
Negative Coefficients in Factoring
- Example: Factor polynomials leading to negative coefficients.
- For the polynomial x² - 11x + 24,
- Identify two numbers that add up to -11 and multiply to 24:
  - The numbers are -3 and -8.
- Hence, the factorization is (x - 3)(x - 8).
Review of Factoring Trinomials
- Students were reminded to approach problems methodically and check work.

Transition to Calculus: This section serves as an introduction to limits and continuity, foundational concepts in calculus.
Definition of a Limit:
- A limit is defined at a point on a coordinate plane when the function approaches a particular value as the input approaches that point from both the left and the right sides.
- If both approaches yield the same value, then the limit exists at that point.
Visual Representation:
- Observe the behavior of a graph as it approaches a certain point:
- Example: To determine the limit approaching the point x = 3:
  - Approach using values less than 3 (e.g., 2.5, 2.7, 2.9) and values greater than 3 (e.g., 3.1, 3.4).
  - If both sides approach the same y-value, the limit exists.
Notation:
- Using notation to express left-hand and right-hand limits:
- Left-hand limit is denoted as $ext{lim}_{x o 3^-} f(x)$ .
- Right-hand limit is denoted as $ext{lim}_{x o 3^+} f(x)$ .
Example of Limits:
- If $ext{lim}_{x o 3} f(x) = 10$ , this means that as x approaches 3 from both directions, the values of f(x) approach 10.

Polynomial Functions: The example given was a simple polynomial $2x + 4$ , which is continuous and has a limit as x approaches any value.
- By substituting x with that value in the polynomial, the limit can be calculated.
Continuous Nature:
- Polynomials do not have breaks and hence possess limits at all points:
- Example: For $x = 3$ , substitute to find the limit: $f(3) = 10$ .

Continuity: A function is continuous at a point if:
- The limit as you approach that point from the left and the right are the same.
- No holes or jumps exist in the graph at that point.
Discontinuous Graphs:
- If a graph jumps or has a hole at a point, it is discontinuous at that point.
- Visual examples of continuous and discontinuous graphs were discussed.
Definition of Gaps and Jumps:
- The presence of vertical asymptotes indicates limits, while horizontal asymptotes indicate behaviors as x approaches infinity or negative infinity.
Limit Notations in Context:
- If two functions have limits, their sums, differences, products, and quotients also have limits, thus preserving the continuity.

Students were encouraged to consider simplifying expressions before plugging in values to determine limits.
The document provided practical examples for further understanding.
Students were reminded of the necessity to work collaboratively on factoring and simplifying polynomial functions for their class handouts and projects.