1.7 Selecting Procedures for Determining Limits

• What is the definition of a limit in calculus?

  The value a function approaches
• One of the basic properties of limits involves the sum
• What is the method to evaluate limits by replacing the variable x with the value it approaches called?

  Direct substitution
• Limit theorems can be applied to sums, differences, products, and quotients
• Which algebraic technique is used to evaluate limits by canceling common factors?

  Factoring
• Limits involving trigonometric functions often require the application of the Squeeze Theorem
• What is the name of the rule used to evaluate limits of indeterminate forms?

  L'Hôpital's Rule
• Infinite limits occur when the function approaches infinity as x approaches a finite value.
• What is a point of discontinuity in a function?

  A point where continuity fails
• The Intermediate Value Theorem requires that the function is continuous
• Match the basic limit property with its formula:

  Sum/Difference ↔️ $\lim_{x \to c} [f(x) \pm g(x)] =$$\lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$

  Constant Multiplication ↔️ $\lim_{x \to c} k \cdot f(x) =$$k \cdot \lim_{x \to c} f(x)$

  Product ↔️ $\lim_{x \to c} [f(x) \cdot g(x)] =$$\lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

  Quotient ↔️ $\lim_{x \to c} \frac{f(x)}{g(x)} =$$\frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$

  Power ↔️ $\lim_{x \to c} [f(x)]^{n} =$$[\lim_{x \to c} f(x)]^{n}$
• What is the condition for evaluating a limit using direct substitution?

  The result must be a real number
• Steps to evaluate a limit using direct substitution:

  1️⃣ State the function and the limit point

  2️⃣ Substitute the value directly into the function

  3️⃣ Simplify to obtain the limit
• What is the limit of 3x^{2} - 5x + 2</latex> as $x$ approaches 2?


  4
• What is the limit of $x +$$2$ as $x$ approaches 3?


  5
• The product property of limits states that \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)</latex>, provided both limits exist
• Match the limit property with its example:

  Sum/Difference ↔️ $\lim_{x \to 2} (x^{2} +$$3x) =$$10$

  Constant Multiplication ↔️ $\lim_{x \to 1} 5x^{3} =$$5$

  Product ↔️ $\lim_{x \to 3} (x \cdot \sin x) =$$3 \sin 3$

  Quotient ↔️ $\lim_{x \to 4} \frac{x^{2}}{x + 1} =$$\frac{16}{5}$

  Power ↔️ $\lim_{x \to 0} (x + 2)^{3} =$$8$
• Direct substitution is always the first method to try when evaluating a limit.
• What is the Constant Multiplication property of limits?

  $\lim_{x \to c} k \cdot f(x) =$$k \cdot \lim_{x \to c} f(x)$
• The Product property of limits states that \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)</latex>, where the hidden word is product
• The Quotient property of limits requires that $\lim_{x \to c} g(x) \neq 0$ to be valid.
• What is the Power property of limits?

  $\lim_{x \to c} [f(x)]^{n} =$$[\lim_{x \to c} f(x)]^{n}$
• Steps to evaluate limits using direct substitution

  1️⃣ State the function and the limit point

  2️⃣ Substitute the value directly into the function

  3️⃣ Simplify to obtain the limit
• $\lim_{x \to 2} (3x^{2} - 5x +$$2) =$$4$ demonstrates direct substitution
• The Sum and Difference theorems of limits state that \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)</latex>
• What is the condition for the Quotient theorem of limits to be valid?

  $\lim_{x \to c} g(x) \neq 0$
• Steps to evaluate limits using factoring and simplifying

  1️⃣ Factorize numerator and denominator

  2️⃣ Simplify the expression

  3️⃣ Apply direct substitution
• What is the definition of a limit?

  Value the function approaches
• The basic properties of limits are used to simplify the process of evaluating complex functions.
• Match the property of limits with its formula:

  Sum/Difference ↔️ $\lim_{x \to c} [f(x) \pm g(x)] =$$\lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$

  Constant Multiplication ↔️ $\lim_{x \to c} k \cdot f(x) =$$k \cdot \lim_{x \to c} f(x)$

  Product ↔️ $\lim_{x \to c} [f(x) \cdot g(x)] =$$\lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

  Quotient ↔️ $\lim_{x \to c} \frac{f(x)}{g(x)} =$$\frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$

  Power ↔️ $\lim_{x \to c} [f(x)]^{n} =$$[\lim_{x \to c} f(x)]^{n}$
• What does it mean to evaluate a limit using direct substitution?

  Replace x with its value
• If the result of direct substitution is a real number, the limit is that value.
• Steps to evaluate limits using direct substitution

  1️⃣ State the function and the limit point

  2️⃣ Substitute the value directly into the function

  3️⃣ Simplify to obtain the limit
• To evaluate limits using direct substitution, replace the variable x
• What is the first step in evaluating a limit using direct substitution?

  State the function and the limit point
• Direct substitution always results in a real number as the limit.

  False
• When evaluating limits using direct substitution, you replace the variable x
• What do you do if the result of direct substitution is a real number?

  The limit is that value
• Steps to evaluate limits using direct substitution

  1️⃣ State the function and the limit point

  2️⃣ Substitute the value directly into the function

  3️⃣ Simplify to obtain the limit
• Direct substitution is the only method to evaluate limits.

  False