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What is the factored form of $a^2 - b^2$?
$(a + b)(a - b)$
What is the factored form of $a^3 - b^3$?
$(a - b)(a^2 + ab + b^2)$
What is the factored form of $a^3 + b^3$?
$(a + b)(a^2 - ab + b^2)$
Explain the concept of factoring in the context of limits.
Factoring simplifies expressions by breaking them down into smaller parts, revealing cancellations that allow for direct substitution to evaluate limits.
Why is factoring important when evaluating limits of rational functions?
Factoring helps simplify rational functions, potentially removing indeterminate forms like 0/0, which allows for the evaluation of the limit.
How do you approach finding the limit of a rational function as x approaches a value, and direct substitution results in an indeterminate form?
1. Factor both the numerator and denominator. 2. Cancel any common factors. 3. Substitute the value of x into the simplified expression. 4. If the denominator is still zero, the limit may not exist; further investigation is needed.
What is the first step when evaluating a limit of a polynomial?
Attempt direct substitution. If it results in a defined value, that is the limit. If not, consider algebraic manipulation like factoring.
How do you determine if a limit does not exist after factoring and simplifying?
If, after simplifying, substituting the value x approaches results in division by zero, the limit likely does not exist. Further analysis (e.g., one-sided limits) might be needed to confirm.