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Discontinuity Classification

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About This MicroSim

This MicroSim helps you learn to classify the four types of discontinuities by analyzing limit behavior at specific points:

  1. Removable Discontinuity: The limit exists (both one-sided limits are equal), but either f(a) is undefined or f(a) differs from the limit. Graphically, this appears as a "hole" in the graph.

  2. Jump Discontinuity: Both one-sided limits exist, but they are not equal. The function "jumps" from one value to another at the point of discontinuity.

  3. Infinite Discontinuity: The function approaches positive or negative infinity as x approaches the point of discontinuity. This creates a vertical asymptote.

  4. Essential Discontinuity: The limit does not exist because the function oscillates infinitely or behaves chaotically near the point. Neither one-sided limit exists.

Two Modes

  • Gallery View: Study reference examples of all four discontinuity types side by side. Each panel shows the graph with the one-sided limit values displayed.

  • Quiz Mode: Test your classification skills. A random function with a discontinuity is displayed, and you must identify which type it is. Immediate feedback explains why your answer is correct or incorrect.

Embedding This MicroSim

You can include this MicroSim on your website using the following iframe:

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<iframe src="https://dmccreary.github.io/calculus/sims/discontinuity-types/main.html" height="502px" scrolling="no"></iframe>

Lesson Plan

Learning Objective

Students will classify discontinuities by analyzing limit behavior at specific points.

Bloom's Taxonomy Level: Analyze (L4)

Bloom's Verb: Classify

Grade Level

High School AP Calculus (Grades 11-12)

Prerequisites

  • Understanding of limits and limit notation
  • Knowledge of one-sided limits
  • Familiarity with function graphs
  • Understanding of what it means for a limit to exist or not exist

Duration

15-20 minutes

Activity Sequence

  1. Introduction (3 minutes)
  2. Open the MicroSim in Gallery View
  3. Review the four discontinuity types with the class

  4. Point out the key distinguishing features in each panel

  5. Guided Exploration (5 minutes)

  6. Click each panel to highlight it
  7. Discuss the one-sided limit values shown

  8. Ask students: "What makes each type different?"

  9. Independent Practice (7-10 minutes)

  10. Have students switch to Quiz Mode
  11. Students work through at least 5-6 quiz questions

  12. Encourage students to read the explanations for each answer

  13. Wrap-up Discussion (2-3 minutes)

  14. Ask students to articulate the decision process for classification
  15. Create a class flowchart: "How do I classify a discontinuity?"

Assessment Questions

  1. If both one-sided limits equal 5, but f(2) = 3, what type of discontinuity is at x = 2?

  2. At x = 1, the left-hand limit is 4 and the right-hand limit is 7. What type of discontinuity is this?

  3. If a function has a vertical asymptote at x = 3, what type of discontinuity does it have there?

  4. The function sin(1/x) oscillates infinitely as x approaches 0. What type of discontinuity is at x = 0?

Extension Activities

  • Have students create their own examples of each discontinuity type
  • Challenge students to write piecewise functions that produce each type
  • Discuss real-world situations that model each discontinuity type

References