Lesson 7: 4-7 L'Hospital Rule
Duration of Days: 1
Lesson Objective
Apply L’Hospital’s Rule by taking the derivatives of the numerator and denominator separately to evaluate a limit.
Understand that L’Hospital’s Rule is not the Quotient Rule; it is a specific limit evaluation technique.
Use the rule repeatedly for problems where the first application still results in an indeterminate form.
Justify the use of the rule by clearly stating that the individual limits of the numerator and denominator both approach 0 or infinity.
Why does the ratio of the rates of change (derivatives) help us find the value of a function at an "undefined" hole?
What happens if we apply L’Hospital’s Rule to a limit that is not indeterminate? (Answer: You get a wrong answer!)
How many times can you apply L’Hospital’s Rule to a single problem?
How does the College Board expect us to "show work" to prove the limit is indeterminate?
L’Hospital’s Rule
Indeterminate Form
Differentiability (as a requirement)
Continuity (as a requirement)
Quotient Rule (as a "non-example")
Transcendental Functions (often used in these limits)
Topic 4.7: Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms.
Standard LIM-4.A: Determine limits of functions that result in indeterminate forms.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Description:
The lesson begins by revisiting limits from Unit 1 that were solved by factoring or conjugates. Students then learn the definition and formula for L'Hospital Rule.
Purpose: This rule is essential for evaluating limits of "mixed" functions (e.g., a polynomial over an exponential) where algebraic manipulation is impossible. On the AP Exam, it often appears in Free Response questions where students must formally show the limits of the numerator and denominator separately before applying the rule.
DOK 1 (Recall): Identifying if a limit is indeterminate.
DOK 2 (Skill/Concept): Correcting a common error where a student accidentally uses the Quotient Rule instead of L’Hospital’s.
DOK 3 (Strategic Thinking): Determining how to evaluate a limit that requires two or three successive applications of the rule.
Scaffolding (Support):The Notation Police: Use a checklist to ensure students don't write 0/0." (The College Board considers 0/0 to be an "illegal" mathematical statement). Instead, have them write: lim f(x) = 0 and lim g(x) = 0.
Color-Coding: Highlight the numerator in one color and the denominator in another to remind students to differentiate them independently.
AP College Board Assessments