How can a known derivative evaluate a limit?
You will be able to: Recognize a derivative definition inside a limit expression.
How can a known derivative evaluate a limit?
A complicated-looking limit can be a familiar local slope written out in full. Identifying the function and the target input can save unnecessary algebra.
A useful starting point: What is special about eˣ and ln x? →
Words and symbols before equations
- Derivative pattern
- [f(a+h)−f(a)]/h as h→0, or [f(x)−f(a)]/(x−a) as x→a.
- Base point
- The fixed input a in the difference quotient.
- Increment
- The difference from the base input.
- Recognition
- Matching every part of an expression to a known definition.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. For logarithm, step is 0.1×10⁻ᵖ so both sides remain positive; other steps are 10⁻ᵖ. Stable algebraic identities avoid subtracting nearly equal values. Limits are not evaluated by dividing by zero.
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.
- The expression [f(a+h)−f(a)]/h tends to f′(a). The current value is one sample, not the limit.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The limit lim h→0 [sin(π/3+h)−sin(π/3)]/h matches the derivative of sine at a=π/3. Its value is cos(π/3)=1/2.
The limit lim x→1 [ln x−ln 1]/(x−1) is the derivative of ln at 1, so it is 1. The subtraction ln 1=0 may be invisible after simplification.
A denominator that is a constant multiple of the input increment changes the result. For [e^(2+h)−e²]/(2h), factor out 1/2; the limit is e²/2.
This is use of the derivative definition after the rules are known, not L’Hôpital’s rule. Check that the numerator subtracts the function’s value at the same base point.
A worked example, step by step
Evaluate lim x→0 (cos x−1)/x.
- Choose f(x)=cos x and a=0.
- The subtracted value 1 equals f(0).
- The denominator x is x−0, so the expression is a derivative quotient.
- The limit is f′(0)=−sin 0=0.
Match the entire denominator to the increment; an extra factor does not disappear.
Is lim h→0 [e^(1+h)−e]/(3h) equal to e?
Compare with an explanation
No. It is one third of the derivative at 1, hence e/3.
Predict. Change one thing. Explain.
Choose sine, exponential or logarithm at the stated base point. Shrink h from both sides and compare the numerical quotient with the exact derivative shown.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The expression [f(a+h)−f(a)]/h tends to f′(a). The current value is one sample, not the limit.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. For logarithm, step is 0.1×10⁻ᵖ so both sides remain positive; other steps are 10⁻ᵖ. Stable algebraic identities avoid subtracting nearly equal values. Limits are not evaluated by dividing by zero.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain 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 h→0 [sin(π/6+h)−1/2]/(2h) by identifying the function, point and scale factor.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The function is sine and the base point is π/6.
- 1 point: sin(π/6)=1/2, matching the subtraction.
- 1 point: Factor out 1/2 from the denominator.
- 1 point: The limit is (1/2)cos(π/6)=sqrt(3)/4 in radians.
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 1What should the subtraction equal?
f(a) at the same base point.
RECALL 2What if the denominator is 5h?
The derivative value is multiplied by 1/5.
RECALL 3Is this L’Hôpital’s rule?
No. It directly recognizes the derivative definition.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can a known derivative evaluate a limit?
- A matching difference-quotient limit equals f′(a).
- Constant multipliers outside a quotient remain in the limit.
Remember: Match the entire denominator to the increment; an extra factor does not disappear.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. For logarithm, step is 0.1×10⁻ᵖ so both sides remain positive; other steps are 10⁻ᵖ. Stable algebraic identities avoid subtracting nearly equal values. Limits are not evaluated by dividing by zero.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.A, LIM-3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.7, FUN-3.A, LIM-3.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and 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 3.1, 3.3 and 3.5 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 uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
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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