Case File Teacher GuideCase File: Teacher Guide Lesson: Limit Lawsuit (Precalculus) Objective Students will calculate overall limits from graphs and determine when limits **Do Not Exist (DNE)** by evaluating the agreement between left-hand and right-hand limits. They will categorize discontinuities (removable, jump, infinite, oscillating) and understand their impact on limit existence. At a Glance 60 Minutes Individual & Pairs 11th-12th Grade Lesson Timeline 0:00-0:05 Warm-up: The Summons Review left-hand \(\lim_{x \to a^-} f(x)\) and right-hand \(\lim_{x \to a^+} f(x)\) notation. Students use the warm-up section on their worksheet to identify limits from a basic piecewise graph. 0:05-0:20 Direct Instruction: Evidence Review Watch the video "Limits and Discontinuities". 0:00 - 2:20: The definition of a limit (LHL = RHL). 10:25 - 13:00: Guided practice. Pause at 10:43 to let students attempt the limits for \(f(x)\) and \(h(x)\) on their worksheet before the narrator explains. 0:20-0:35 Simulation: The Limit Lawyer Pair activity using the Limit Lawyer Cards. One student is the "Left Hand" lawyer, the other "Right Hand". They check specific x-values on complex graphs. If they agree on the value, the limit "passes". If not, the case is "thrown out" (DNE). 0:35-0:50 Application: Legal Limits Worksheet Students work independently on the Legal Limits Worksheet, identifying limits and discontinuities from various graphs and justifying DNE responses. 0:50-1:00 Closing: Closing Arguments Journal prompt: "Why can a limit exist where a function value does not?" Encourage students to use the term "Removable Discontinuity" or "Hole" in their explanation. Defense Strategies (Avoid Misconceptions) "The function value is undefined, so the limit is DNE." Remind students that limits are about the approach, not the arrival. Holes (removable discontinuities) are the classic case where the limit exists but the function value does not. "Both sides go to \(\infty\), so the limit exists." Clarify that strictly speaking, \(\infty\) is not a number. The limit DNE because it doesn't settle on a finite value. We use \(\lim = \infty\) only to describe the behavior of the non-existent limit.
Limit Lawsuit SlidesLimit Lawsuit Evidence-Based Calculus Foundations The Summons Before we begin the trial, review your legal notation for One-Sided Limits. \(\lim_{x \to a^-} f(x)\) Left-Hand \(\lim_{x \to a^+} f(x)\) Right-Hand Left Approach Right Approach The Full Definition A function \(f(x)\) has an overall limit \(L\) at \(x = a\) if and only if: \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\) If the one-sided limits disagree, the overall limit Does Not Exist (DNE). Evidence: Intro Timestamps: 0:00 - 2:20 Embedded media Identifying the Overall Limit and conditions for failure. Types of Discontinuities Removable A "hole" in the graph. The limit exists because paths match! ○ Jump A physical gap. LHL \(\neq\) RHL. The limit is DNE. Infinite A vertical asymptote. The limit is DNE or \(\infty\). Cross-Examination Timestamps: 10:25 - 13:00 Embedded media Pause at 10:43! Check your worksheet. Attempt the limits for \(f(x)\) and \(h(x)\) on your own first. Find the holes, jumps, and asymptotes. The Limit Lawyer Partner A is the Left-Hand Lawyer. Partner B is the Right-Hand Lawyer. Evaluate your side of the approach for the case file. If you agree on the value, the Limit Law Passes. If you disagree, the Case is Thrown Out (DNE). Closing Arguments "Why can a limit exist where a function value does not?" Write your response in your reflection journal.
Legal Limits WorksheetLegal Limits Evidence Log Case ID: CALC-LIMIT-01 // Precalculus Division Name: __________________________ Date: __________________________ Exhibit A: The Summons Study the preliminary evidence (the graph below) and provide the requested one-sided limits. 1. \(\lim_{x \to -2^-} f(x) =\) 2. \(\lim_{x \to -2^+} f(x) =\) 3. \(f(-2) =\) The Verdict Question: Based on the definition of an overall limit, does the limit exist at \(x = -2\)? Justify your answer using the data from above. Exhibit B: Cross-Examination Practice Pause the video at 10:43. Record your independent findings for each case below before the trial continues. Trial 1: Function \(f(x)\) \(\lim_{x \to -3} f(x) =\) \(\lim_{x \to -2} f(x) =\) \(\lim_{x \to 0} f(x) =\) Trial 2: Function \(h(x)\) \(\lim_{x \to -1} h(x) =\) \(\lim_{x \to 0} h(x) =\) \(\lim_{x \to 3} h(x) =\) Exhibit C: Discontinuity Classification Identify the impact of each "incident type" on the existence of an overall limit. Incident TypeVisual DescriptionDoes the Limit Exist?Removable DiscontinuityA single "hole" in the graph; the path remains predictable from both sides.YES / NOJump DiscontinuityA physical gap where the left and right paths lead to different heights.YES / NOInfinite DiscontinuityA vertical asymptote where the function values grow without bound.YES / NOOscillating DiscontinuityThe function values vibrate wildly and never settle as they approach.YES / NO Exhibit D: Final Summation A lawyer discovers a function \(g(x)\) where the evidence shows \(\lim_{x \to 5^-} g(x) = 12\) and \(\lim_{x \to 5^+} g(x) = 12\). However, witness testimony confirms \(g(5)\) is undefined. Provide the final verdict for \(\lim_{x \to 5} g(x)\) and explain your reasoning based on the "Full Definition" of a limit.
Limit Lawyer CardsLimit Lawyer Activity Cards Cut along the dotted lines. Use in pairs: one Left Hand Lawyer, one Right Hand Lawyer. Subpoena: Case #101 Evaluate Limit at \(x = 0\) Left-Hand Counsel: Verdict: _________ Right-Hand Counsel: Verdict: _________ Judge's Final Verdict (Overall Limit): _________ Subpoena: Case #102 Evaluate Limit at \(x = 1\) Left-Hand Counsel: Verdict: _________ Right-Hand Counsel: Verdict: _________ Judge's Final Verdict (Overall Limit): _________ Subpoena: Case #103 Evaluate Limit at \(x = -1\) Left-Hand Counsel: Verdict: _________ Right-Hand Counsel: Verdict: _________ Judge's Final Verdict (Overall Limit): _________ Subpoena: Case #104 Evaluate Limit at \(x = 0\) Left-Hand Counsel: Verdict: _________ Right-Hand Counsel: Verdict: _________ Judge's Final Verdict (Overall Limit): _________ Cut along solid black borders. Ensure each pair has one of each case file.
Closing Arguments ReflectionClosing Arguments: Reflection Journal Official Inquiry #1 Based on today's courtroom evidence, explain in your own words: Why can a limit exist at a point where the function value does not? Use the term "approach" in your explanation. Official Inquiry #2 Consider a Jump Discontinuity. Why must the "case be thrown out" (DNE) for the overall limit, even if both the left-hand lawyer and right-hand lawyer have valid values? Final Statement "A limit is a prediction based on path, not a destination." Do you agree with this legal precedent? Why?
