1.6 Determining Limits Using Algebraic Manipulation

• 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?

  $a^{2} - b^{2} =$$(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?

  $a^{2} - b^{2} =$$(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, $lim_{x→a^ - } f(x) =$$L$

  Limit from the Right ↔️ Value f(x) approaches as x approaches a from the right side, $lim_{x→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 $lim_{x→1} (√x - 1) / (x - 1)$, 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?

  $a^{2} - b^{2} =$$(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