Limits, Function Evaluation, and Polynomial Factoring

Question 1: Simplifying Expressions and Negative Exponents

This section focuses on simplifying algebraic expressions and rewriting them without negative exponents, a foundational skill for calculus involving functions and their properties.

• Negative Exponents: Recall that for any nonzero number and integer .

• Simplification: Combine like terms and use exponent rules to rewrite expressions with only positive exponents.

• Example:

Question 2: Factoring Polynomials

Factoring polynomials into products of first-degree (linear) factors is essential for solving equations and analyzing functions in calculus.

• Factoring: Express a polynomial as a product of linear factors with integer coefficients.

• Example:

• Application: Factoring is used to find roots and simplify rational expressions.

Question 2 (continued): Estimating Limits and Function Values from a Graph

Limits describe the behavior of a function as the input approaches a particular value. Graphical analysis is a key method for estimating limits and function values in calculus.

• Limit Notation: denotes the value that approaches as approaches .

• Function Value: is the actual value of the function at .

• Example (from graph):

  • If left and right limits differ, does not exist (DNE).

• Additional info: When the graph has a jump or discontinuity at , the limit may not exist even if is defined.

Question 3: Evaluating Limits Algebraically

Algebraic techniques are used to evaluate limits, especially when direct substitution leads to indeterminate forms or undefined expressions.

• Direct Substitution: If is continuous at , then .

• Indeterminate Forms: If substitution yields or (where ), further simplification or factoring is needed.

• Example:

  • Let

  • a.

  • b. (DNE)

  • c.

• Additional info: If the denominator approaches zero but the numerator does not, the limit does not exist (DNE) due to a vertical asymptote.

Summary Table: Limit Existence and Function Values

The following table summarizes the possible relationships between limits and function values at a point.