When should L’Hôpital’s rule be applied again?
You will be able to: Recheck the limiting form at each step and stop when ordinary limit evaluation works.
When should L’Hôpital’s rule be applied again?
The numerator x² and denominator eˣ both grow without bound as x grows. Their ratio depends on which grows faster, so the symbol ∞/∞ alone does not determine a value.
A useful starting point: When is L’Hôpital’s rule allowed? →
Words and symbols before equations
- Repeated application
- Using the theorem again on a new eligible quotient.
- End behavior
- What happens as the input grows without bound.
- Extended limit
- A statement of unbounded growth toward +∞ or −∞, not an ordinary real value.
- Recheck
- Verify the form and differentiability conditions anew before another application.
What this picture assumes
Original mathematical model; readouts are rounded. Both examples require two justified derivative-ratio steps. Near-zero uses nonzero x=10^(−p); large-input uses its own slider. Only the selected example’s relevant control changes its values. Stable sine identities avoid cancellation at small inputs.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Selected x=0.1: blue original=0.499583, orange first ratio=0.499167, teal second ratio=0.497502. Common limit 0.5. Recheck the indeterminate form before each step; the ratios are not identical functions.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For lim x→∞ x²/eˣ, the first form is ∞/∞. Differentiating the components gives 2x/eˣ, which is still ∞/∞.
A second eligible application gives 2/eˣ, whose limit is 0 directly. Thus the preceding quotient limit is 0, and the first application then gives the original limit 0.
For lim x→0 (1−cos x)/x², the first derivative ratio is sin x/(2x), still 0/0. Differentiating again gives cos x/2→1/2. The required denominator derivatives are nonzero at sufficiently close nonzero x for the first application and everywhere for the second.
Stop once the new limit is determined. Continuing to differentiate a quotient that is no longer indeterminate is not justified by this rule. The method concerns limiting comparison, not repeatedly changing functions until an answer looks simple.
A worked example, step by step
Evaluate lim x→0 (eˣ−1−x)/x².
- The original form is 0/0; the functions are differentiable nearby and 2x≠0 for x≠0.
- First derivative ratio: (eˣ−1)/(2x), again 0/0.
- Second derivative ratio: eˣ/2, with nonzero denominator derivative 2 at that second application.
- Its limit is 1/2, establishing the two earlier quotient limits in reverse order.
A second application needs a fresh 0/0 or ∞/∞ check. A determinate form should be evaluated directly.
Why is 2/eˣ not another ∞/∞ form?
Compare with an explanation
Its numerator stays 2 while its denominator grows, so ordinary limit reasoning gives zero.
Predict. Change one thing. Explain.
Switch between the near-zero and large-input cases. Compare the original quotient and each derivative ratio without treating their finite-input values as identical.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Selected x=0.1: blue original=0.499583, orange first ratio=0.499167, teal second ratio=0.497502. Common limit 0.5. Recheck the indeterminate form before each step; the ratios are not identical functions.
Original mathematical model; readouts are rounded. Both examples require two justified derivative-ratio steps. Near-zero uses nonzero x=10^(−p); large-input uses its own slider. Only the selected example’s relevant control changes its values. Stable sine identities avoid cancellation at small inputs.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant rate relationship, tangent estimate, or limit argument and its conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionEvaluate lim x→∞ x/eˣ and state why only one L’Hôpital step is needed.
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Compare with the answer and four-point rubric
- 1 point: The form is ∞/∞.
- 1 point: The functions are differentiable for large x and denominator derivative eˣ is nonzero.
- 1 point: The derivative ratio is 1/eˣ.
- 1 point: Its limit is 0 directly; no second application is needed.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why can repetition be needed?
The first derivative ratio may still have an indeterminate quotient form.
RECALL 2When do you stop?
When the limit is evaluable by ordinary rules or a different method is needed.
RECALL 3Must finite-input ratios match?
No; the theorem compares their limits under its assumptions.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When should L’Hôpital’s rule be applied again?
- Recheck the form and hypotheses at every application.
- lim x→∞ x²/eˣ=0; lim x→0 (1−cos x)/x²=1/2.
Remember: A second application needs a fresh 0/0 or ∞/∞ check. A determinate form should be evaluated directly.
Conditions: Original mathematical model; readouts are rounded. Both examples require two justified derivative-ratio steps. Near-zero uses nonzero x=10^(−p); large-input uses its own slider. Only the selected example’s relevant control changes its values. Stable sine identities avoid cancellation at small inputs.
Refresh Kid · AP Calculus AB Unit 4 · Objectives LIM-4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.7, LIM-4.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 4 has seven official topics. Topic 4.7 assesses 0/0 and ∞/∞ quotient forms; other indeterminate forms are excluded from the core lessons. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative units and signs are interpreted in context. Speed is the magnitude of velocity; turning requires a sign change. Related-rate equations hold at nearby times and are differentiated before snapshot values are inserted. Cone and ladder models have explicit physical domains. Tangent approximations remain estimates; error direction requires behavior on the relevant interval. L’Hôpital’s rule requires an eligible quotient form, nearby differentiability, nonzero denominator derivative and an existing finite or infinite derivative-ratio limit. A failed derivative-ratio limit is inconclusive about the original quotient.
The Organic Chemistry Tutor video creators and relevant descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.4, 4.1, 4.2 and 4.8 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model shows the circular water surface and axial cross-section of a tip-down conical tank. The water radius and height obey the same similar-triangle ratio in 2D and 3D. Camera rotation only changes the view; signed flow and height controls describe an instantaneous state. Complete labeled 2D geometry, rates and equations remain available without WebGL.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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