Can the limit differ from the value at the point?
You will be able to: Separate nearby behavior from an assigned point value.
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.
Can the limit differ from the value at the point?
A graph follows y=x+2 near x=1, but a single filled dot at x=1 is placed at height 5. Nearby outputs head toward 3 even though the assigned output is 5.
A useful starting point: What does a limit statement actually say? →
Words and symbols before equations
- f(a)
- The output assigned exactly at input a.
- Open circle
- A coordinate excluded from that graph piece.
- Filled point
- An included function value.
- Removable discontinuity
- A point mismatch or hole with a finite common limit.
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)=x+2 for x≠1. The slider independently assigns f(1). Open circle excludes a branch point; filled point shows the assigned value.
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.
- The left and right limits equal 3; f(1)=5. 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 mathematics
To find the limit, follow the curve at inputs just less and just greater than 1. Both approach height 3. The isolated filled point is not nearby behavior.
To find f(1), locate the included point exactly above input 1. Its height is 5. Thus the limit is 3 while f(1)=5.
Changing only f(1) leaves every other input unchanged, so it cannot change this limit. Removing f(1) entirely also leaves the limit at 3.
Continuity later asks whether the assigned value agrees with the nearby behavior. Having a limit alone does not guarantee that agreement.
| Question | Limit as x approaches a | Value f(a) |
|---|---|---|
| Where to look | Inputs near a, on both sides | Exactly x=a |
| Graph marker | The approached height | A filled point, when defined |
| Change only f(a) | Limit is unchanged | Assigned value changes |
A worked example, step by step
Let f(x)=x+2 for x≠1 and f(1)=−1. Find the limit at 1 and the value at 1.
- For every nearby x≠1 use x+2.
- As x→1, x+2→3 from both sides.
- The special definition says f(1)=−1.
- The limit is 3, the value is −1; they are different questions.
An open circle does not automatically mean the limit fails to exist.
What value would make this graph continuous at 1?
Compare with an explanation
Set f(1)=3, the common nearby limit.
Predict. Change one thing. Explain.
Move the assigned value at x=1. Keep the surrounding line fixed. Identify the one setting where value and limit agree.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The left and right limits equal 3; f(1)=5. 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)=x+2 for x≠1. The slider independently assigns f(1). Open circle excludes a branch point; filled point shows the assigned value.
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 f(x)=2x−1 when x≠2 and f(2)=7, determine both quantities at 2, describe the open and filled points, and repair continuity.
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Compare with the answer and four-point rubric
- 1 point: The nearby limit is 3.
- 1 point: f(2)=7.
- 1 point: Show an open point at (2,3) and a filled point at (2,7).
- 1 point: Redefine f(2)=3 to match the limit.
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 1Does a limit determine f(a)?
Not unless more information, such as continuity, is given.
RECALL 2What does a filled dot specify?
The included value at that input.
RECALL 3Can an undefined f(a) coexist with a limit?
Yes.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Can the limit differ from the value at the point?
- Changing one assigned point does not change a limit determined by nearby inputs.
Remember: An open circle does not automatically mean the limit fails to exist.
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)=x+2 for x≠1. The slider independently assigns f(1). Open circle excludes a branch point; filled point shows the assigned value.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-1.A, LIM-1.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.2, LIM-1.A, LIM-1.B. 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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