What can a table tell you about a limit?
You will be able to: Estimate a limit from both sides and state what the table cannot prove.
BC foundation: Unit 1 shares its limits-and-continuity objectives with AB. If you have studied these ideas before, use the explanations and written challenges to check your reasoning. Later BC units build on these foundations; this unit does not require series or advanced integration.
What can a table tell you about a limit?
A missing measurement at x=3 does not prevent us from looking at measurements near 3. Inputs 2.9 and 3.1 for (x²−9)/(x−3) give outputs near 6.
A useful starting point: Can a graph window hide limit behavior? →
Words and symbols before equations
- Table estimate
- A conjecture supported by sampled input–output pairs.
- Approach distance
- The positive difference between an input and the target.
- Undefined
- No assigned real value for that expression at the input.
- Precision
- How many reliable digits are retained.
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. Original expression (x²−9)/(x−3), x≠3. Stable nearby formula x+3 is used for numeric display; the target remains undefined. Table decimals are rounded.
Read the picture in three steps
- Read the axes, coordinates and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Both sides suggest 6. The identity (x²−9)/(x−3)=x+3 for x≠3 establishes that limit; a finite table alone does not prove it.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Use inputs smaller and larger than the target. For this expression, 2.9, 2.99 and 2.999 give 5.9, 5.99 and 5.999; the right-side values approach 6 from above.
At x=3 the original expression is 0/0, which is undefined. This says nothing yet about the limit.
Factoring gives (x−3)(x+3)/(x−3)=x+3 for x≠3. That exact nearby formula justifies the estimated limit 6.
Very small differences can cause calculator cancellation or rounding. A displayed 0/0 or flat table may reflect numerical limitations; use algebra and state the original domain.
A worked example, step by step
Estimate the limit from values at 2.99 and 3.01, then justify it.
- At 2.99 the output is 5.99.
- At 3.01 the output is 6.01.
- The two sides suggest 6.
- Factoring makes the expression x+3 for x≠3, so the limit is exactly 6.
A finite table is evidence, not a proof about every sufficiently nearby input.
Should we average the last two entries and call that a proof?
Compare with an explanation
No. Such an average is at most an estimate; the nearby formula supplies the justification.
Predict. Change one thing. Explain.
Reduce the table spacing. Compare left and right rows. Explain why the original expression remains undefined at 3 even as the nearby values converge.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Both sides suggest 6. The identity (x²−9)/(x−3)=x+3 for x≠3 establishes that limit; a finite table alone does not prove it.
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. Original expression (x²−9)/(x−3), x≠3. Stable nearby formula x+3 is used for numeric display; the target remains undefined. Table decimals are rounded.
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 (x²−25)/(x−5), give values at 4.9 and 5.1, conjecture the limit, and justify without changing the original domain.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: At 4.9 the value is 9.9.
- 1 point: At 5.1 the value is 10.1.
- 1 point: Conjecture limit 10.
- 1 point: For x≠5 the quotient equals x+5, whose limit is 10; the original remains undefined at 5.
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 use both sides?
They may approach different outputs.
RECALL 2What does 0/0 tell us?
The substitution is indeterminate, not the limit’s value.
RECALL 3How can algebra improve a table estimate?
It establishes behavior for all nearby allowed inputs.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What can a table tell you about a limit?
- Sample both sides, avoid the target if undefined, then justify analytically when possible.
Remember: A finite table is evidence, not a proof about every sufficiently nearby input.
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. Original expression (x²−9)/(x−3), x≠3. Stable nearby formula x+3 is used for numeric display; the target remains undefined. Table decimals are rounded.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-1.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.4, LIM-1.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 BC unit destination was 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 BC 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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