1.9 Connecting Multiple Representations of Limits

• Numerical limits are defined using tables by examining values of f(x)
• If a function is continuous at x = c, the limit exists and equals f(c).


  True
• As x approaches 2, the function f(x) = (x^2 - 4)/(x - 2) approaches 4
• The limit of (x^2 - 4)/(x - 2) as x approaches 2 is 4.

  True
• Match the concept with its description:

  Left-hand limit ↔️ Approaching c from the left

  Right-hand limit ↔️ Approaching c from the right

  Continuity ↔️ f(c) equals the limit

  Graphical limit ↔️ Analyzing behavior on a graph
• The limit of (x^2 - 4)/(x - 2) as x approaches 2 is 4.

  True
• A graphical limit exists if the function is continuous at x = c.

  True
• What technique is used to simplify a function with a common factor in the numerator and denominator?

  Factoring and canceling
• Steps to convert a table of data to an algebraic function representation

  1️⃣ Identify the relationship between x and f(x)

  2️⃣ Select a functional form

  3️⃣ Determine the parameters of the function

  4️⃣ Construct the algebraic expression
• What value does f(x) approach as x approaches 2 from the left in the example given?

  3.999
• Steps to identify graphical limits from function graphs

  1️⃣ Examine the behavior of f(x) as x approaches c from the left

  2️⃣ Examine the behavior of f(x) as x approaches c from the right

  3️⃣ Determine if both left-hand and right-hand limits converge to the same value L

  4️⃣ If yes, then lim_x→c f(x) = L
• For rational expressions, factoring and canceling common factors is a key technique for evaluating algebraic limits


  True
• What is the algebraic expression for the linear relationship in the table where f(0) = 3 and f(1) = 5?

  f(x) = 2x + 3
• As x approaches c from the left, f(x) approaches the value L
• For the function f(x) = (x^2 - 4) / (x - 2), the limit as x approaches 2 is 4
• What numerical method can be used to translate an algebraic limit into numerical data?

  Creating a table
• What is the general algebraic representation of a linear function?

  f(x) = mx + b
• What is the limit of f(x) = (x^2 - 4) / (x - 2) as x approaches 2?

  4
• What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \) approaches 2?

  4
• What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \) approaches 2?

  4
• What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \) approaches 2 after factoring and canceling?

  4
• What is the algebraic expression for a linear function with a slope of 2 and a y-intercept of 3?

  f(x) = 2x + 3
• What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \) approaches 2, based on numerical data?

  4
• The function f(x) = 3^x is an exponential function with a base of 3.


  True
• If f(x) converges to a value L from both sides, the limit exists and equals L
• What is the definition of a numerical limit using tables?

  Left and right limits are equal
• Graphical limits exist if the function approaches the same value from both sides and is continuous at x = c.


  True
• When evaluating numerical limits, the values of f(x) are examined as x approaches c from the left
• If f(c) = L and the left and right-hand limits both equal L, then lim_x→c f(x) = L.

  True
• Under what condition does the limit of a function equal its value at x = c?

  Continuity at x = c
• If the left-hand and right-hand limits both approach the same value L, then the limit exists and lim_x→c f(x) = L
• If the function is defined at the limit value c, you can find the limit by direct substitution
• The slope of a linear function can be calculated as the change in f(x) divided by the change in x
• A graphical limit exists if the function's left-hand and right-hand limits converge to the same value
• If the left-hand and right-hand limits of a function approach the same value L, then the limit exists and is equal to L


  True
• When rationalizing the denominator, you multiply by the conjugate
• To find a limit graphically, you observe the function's behavior as x approaches a value c from both sides


  True
• If the left-hand and right-hand limits both approach the same value L, then the limit exists.


  True
• Match the type of function with its graphical behavior:

  Linear Function ↔️ Straight line

  Quadratic Function ↔️ Parabola

  Exponential Function ↔️ Curve that grows rapidly
• For the function f(x) = (x^2 - 4) / (x - 2), as x approaches 1.99 from the left, f(x) approaches 3.99