How do a formula, table and graph tell the same story?
You will be able to: Translate one limit claim across analytical, graphical, numerical and verbal forms.
How do a formula, table and graph tell the same story?
A formula with a hole, a table with a missing row, and a line with an open point can all describe the same nearby behavior.
A useful starting point: Why do trigonometric limits require radians? →
Words and symbols before equations
- Representation
- A formula, graph, table or verbal description of a function.
- Consistency
- Different forms agree about the same inputs and outputs.
- Analytical justification
- Reasoning from an expression or theorem.
- Point assignment
- A separately defined output at the target.
What this picture assumes
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=(x²−9)/(x−3) for x≠3; f(3) is assigned separately. Table and graph preserve both pieces.
Read the picture in three steps
- Read the species and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- The left and right limits equal 6; f(3)=1. The finite limit exists, but the assigned value differs: removable discontinuity. Open circle: branch exclusion; filled point: assigned value.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the chemistry
Take f(x)=(x²−9)/(x−3) for x≠3 and f(3)=1. Algebra gives x+3 away from 3; that describes a line with an open point at (3,6).
The separate definition adds a filled point at (3,1). A table shows outputs near 6 on both sides, while the exact row x=3 must show 1.
Verbally: the output approaches 6 as the input approaches 3, but the output at 3 is 1. Symbolically write the limit as 6 and f(3)=1 separately.
If a graph or table disagrees, return to the domain and piecewise definition. A rounded graph estimate should not overrule an exact algebraic argument.
A worked example, step by step
Describe the graph and limit of f(x)=x+3 for x≠3 with f(3)=1.
- Draw the line y=x+3 away from x=3.
- Mark an open point at (3,6).
- Mark a filled point at (3,1).
- Both branches approach 6, so the limit is 6 although f(3)=1.
Do not silently fill a hole when translating a simplified expression into a graph.
Which representation justifies the exact nearby limit here?
Compare with an explanation
The equality to x+3 for every nearby x≠3, together with continuity of that line.
Predict. Change one thing. Explain.
Change only the assigned point. Compare the formula description, table row at the target and surrounding graph; explain which information remains fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The left and right limits equal 6; f(3)=1. The finite limit exists, but the assigned value differs: removable discontinuity. Open circle: branch exclusion; filled point: assigned value.
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=(x²−9)/(x−3) for x≠3; f(3) is assigned separately. Table and graph preserve both pieces.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using nearby function values, graph coordinates, or the hypotheses of a limit law or theorem. 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 questionFor g(x)=2x for x≠1 and g(1)=−2, supply an open point, filled point, limit statement and a sentence explaining their consistency.
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Compare with the answer and four-point rubric
- 1 point: Open point at (1,2).
- 1 point: Filled point at (1,−2).
- 1 point: lim as x→1 of g(x)=2.
- 1 point: Nearby values follow 2x while the exact input uses the special assignment.
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 must survive simplification?
The original domain and special point values.
RECALL 2Does a table row at a determine the limit?
No; it gives f(a).
RECALL 3What does consistency require?
The representations describe the same function and conditions.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do a formula, table and graph tell the same story?
- Keep the domain and point assignments when translating between representations.
Remember: Do not silently fill a hole when translating a simplified expression into a graph.
Conditions: Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=(x²−9)/(x−3) for x≠3; f(3) is assigned separately. Table and graph preserve both pieces.
Refresh Kid · AP Calculus AB Unit 1 · Objectives LIM-1.A–E; skill 2.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.9, LIM-1.A–E; skill 2.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 1 has 16 official topics. Focused lesson titles and questions are original Refresh Kid teaching material. Topics 1.7 and 1.9 integrate earlier objectives and skills rather than introducing new numbered learning objectives.
Formal epsilon-delta proofs are not assessed in this unit. L’Hôpital’s rule, derivative rules and differentiation tests belong later in the course and are not used to bypass limit reasoning here. Trigonometric limits use radians. Finite graphs and tables provide evidence, not a proof of all nearby behavior. Infinity describes unbounded behavior, not a number to substitute. The Intermediate Value Theorem requires continuity on a closed interval and an intermediate output; it guarantees existence, not uniqueness.
The Organic Chemistry Tutor video title, creator and description were checked; the full video was not reviewed. Khan Academy’s current course announcement and linked unit destination were checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Section 2.2 was 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. Camera rotation does not add a variable or change slope; use labeled coordinates and axis scales.
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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