Day 11 - Limit Problems - 09.02.14

Bell Ringer
  1. Find all values of c such that f is continuous on (-∞, ∞).

    1. both a and b

    2. none of the above

  2. What is the domain of f(x)?

    1. x ≠ 0

    2. x ≥ -c2

    3. x > -c2

    4. both a and b

    5. none of the above

  3. Find the following limit: given that the function is continuous throughout.

    1. 0

    2. f(0)

    3. f(c)

    4. none of the above

  4. Find the following limit:

    1. -∞

    2. 1

    3. 0

    4. none of the above


Review
  • Finding limits using a table
  • Finding limits using a graph
  • Epsilon-Delta Limit Proofs
  • Limit Properties
    • Dividing Out/Rationalizing Techniques
    • Functions that agree at all but one point
  • One-sided Limits
  • Infinite Limits

Lesson


  1. Find the values of the constants a and b such that



  1. Consider the function

    1. Find the domain of f.

    2. Calculate

    3. Calculate


  1. Determine all values of the constant a such that the following function is continuous for all real numbers.

  1. Let a be a nonzero constant. Prove that if , then Show by means of an example that a must be nonzero.



Exit Ticket

  1. The function f and its graph are shown below:

    1. Calculate the limit of f(x) as x gets closer to 2 from the left side.

    2. Which value is greater?

      1. the limit of f(x) as x goes to 1

      2. f(1)

      3. Explain your answer.

    3. At what value(s) of c on the interval [0, 4] does the limit of f(x) as x goes to c not exist?

      1. Explain your answer.

Lesson Objective(s)
  • How are the concepts in chapter 1 related and applied?

Standard(s)
  • APC.2
    • Define and apply the properties of limits of functions.
      • Limits will be evaluated graphically and algebraically.
        • Includes:
          • ​limits of a constant
          • ​limits of a sum, product, and quotient
          • ​one-sided limits
          • ​limits at infinity, infinite limits, and non-existent limits*
  • APC.3
    • Use limits to define continuity and determine where a function is continuous or discontinuous.
      • Includes:
        • ​continuity in terms of limits
        • continuity at a point and over a closed interval
        • ​application of the Intermediate Value Theorem and the Extreme Value Theorem
        • ​geometric understanding and interpretation of continuity and discontinuity
  • APC.4
    • Investigate asymptotic and unbounded behavior in functions.
      • Includes:
        • describing and understanding asymptotes in terms of graphical behavior and limits involving infinity