What is the limit of a function as its input approaches a particular value?
The value it approaches
What is the limit of a constant c as x approaches a?
c
Under what condition can the quotient rule be applied for limits?
Denominator limit ≠ 0
What is the factorization of the expression x² + x - 6?
(x - 2)(x + 3)
The second step in evaluating a limit using factorization is to factorize
When direct substitution leads to an indeterminate form, factorization can help simplify the function to find the limit.
True
Conjugate multiplication is used when direct substitution results in an indeterminate form like 0/0 or ∞/∞.
True
Multiplying by the conjugate simplifies the expression using the difference of squares formula.
True
After canceling common factors in conjugate multiplication, the limit of (√x - 3) / (x - 9) as x approaches 9 is 1/6
Conjugate multiplication simplifies expressions with square roots that lead to indeterminate forms.
True
Steps to simplify and evaluate a limit using conjugate multiplication
1️⃣ Identify the term with the square root
2️⃣ Multiply the numerator and denominator by the conjugate
3️⃣ Simplify using the difference of squares formula
4️⃣ Cancel common factors
5️⃣ Evaluate the limit by direct substitution
The conjugate of an expression containing a square root is used to simplify limits by multiplying both the numerator and denominator
Match the technique with its description:
Direct Substitution ↔️ Plug the given value into the function
Factoring and Canceling ↔️ Factor and cancel common terms
Rationalizing ↔️ Multiply by a conjugate
The limit of a product is the product of the limits
Steps to apply factorization techniques in limit problems:
1️⃣ Identify common factors, perfect squares, or trinomials
2️⃣ Factorize the numerator and/or denominator
3️⃣ Cancel any common factors
4️⃣ Evaluate the limit by direct substitution
What is the first step in evaluating a limit using factorization techniques?
Identify common factors
The final step in evaluating a limit using factorization is to evaluate
What is the trinomial factorization technique used for?
Breaking down ax² + bx + c
What is the conjugate of √x - 2?
√x + 2
What is the conjugate of √x - 3?
√x + 3
What formula is used to factor the difference of squares?
a2−b2=(a+b)(a−b)
What is the limit of (√x - 2) / (x - 4) as x approaches 4?
1/4
What is the formula for the difference of squares?
a2−b2=(a+b)(a−b)
Steps to evaluate lim x→4 (√x - 2) / (x - 4)
1️⃣ Identify the square root term: √x - 2
2️⃣ Multiply numerator and denominator by the conjugate: (√x - 2) / (x - 4) * (√x + 2) / (√x + 2)
3️⃣ Simplify using the difference of squares: (x - 4) / ((x - 4)(√x + 2))
4️⃣ Cancel common factors: 1 / (√x + 2)
5️⃣ Evaluate by direct substitution: 1 / (√4 + 2) = 1 / 4
Match the limit type with its description and notation:
Limit from the Left ↔️ Value f(x) approaches as x approaches a from the left side, limx→a−f(x)=L
Limit from the Right ↔️ Value f(x) approaches as x approaches a from the right side, limx→a+f(x)=L
The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not equal to zero
Conjugate multiplication is used to simplify limits involving square roots when direct substitution results in an indeterminate form.
True
Conjugate multiplication is used to simplify limits involving square roots, especially when direct substitution results in an indeterminate form like 0/0 or ∞/∞
What is the conjugate of √x - 3?
√x + 3
When using conjugate multiplication for limx→1(√x−1)/(x−1), the conjugate to multiply by is √x + 1
Limits can be one-sided, meaning the limit from the left side and right side may be different.
True
The limit of a sum is the sum of the limits.
True
The limit of a function raised to a power is the limit raised to that power.
True
The trinomial x² - 16 can be factored as (x - 4)(x + 4
What should you do after factoring in a limit problem?
Cancel common factors
What formula is used in the difference of squares factorization technique?
a2−b2=(a+b)(a−b)
The first step in applying conjugate multiplication is to identify
Steps to apply conjugate multiplication
1️⃣ Identify the term with the square root
2️⃣ Multiply the numerator and denominator by the conjugate
3️⃣ Simplify using the difference of squares formula
4️⃣ Cancel common factors
5️⃣ Evaluate the limit by direct substitution
Why is factoring and canceling important when simplifying rational expressions in limits?
Eliminates indeterminate forms
To simplify (√x - 2) / (x - 4), you multiply both numerator and denominator by √x + 2