How do you read a graph from each side?
You will be able to: Use one-sided limits to decide whether a two-sided limit exists.
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.
How do you read a graph from each side?
A delivery charge jumps at a weight threshold. Just below the threshold costs can approach one amount while just above it they approach another.
A useful starting point: Can the limit differ from the value at the point? →
Words and symbols before equations
- Left-hand limit
- Outputs approached from inputs smaller than the target.
- Right-hand limit
- Outputs approached from inputs larger than the target.
- a⁻ and a⁺
- Approach directions in the input, not signs of the output.
- Two-sided limit
- The shared finite value of both one-sided limits, when equal.
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. For x<1 use x+1; for x>1 use the selected branch; f(1)=6. Curves are split at the excluded endpoints, never joined across the break.
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.
- Left limit=2; right limit=4; f(1)=6. Side limits differ, so the two-sided limit does not exist.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Trace the graph toward x=1 from the left and read its vertical height. Then repeat from the right. Never decide which side by whether the output is positive or negative.
For x<1 use f(x)=x+1; for x>1 use f(x)=x+3. The respective limiting heights are 2 and 4.
Since 2≠4, no single finite value describes both sides. The two-sided limit does not exist even if f(1) is assigned a value.
If the right branch changes to x+1, both sides approach 2 and the limit exists. The point value remains a separate issue.
A worked example, step by step
For f(x)=x+1 for x<1, f(1)=6 and f(x)=x+3 for x>1, find all three values and the two-sided limit.
- Left side: x+1→2.
- Right side: x+3→4.
- The filled point gives f(1)=6.
- The left and right limits differ, so the two-sided limit does not exist.
The superscript minus in 1⁻ means inputs below 1, not negative outputs.
Can choosing a different f(1) repair a jump?
Compare with an explanation
No. It does not change the unequal side limits.
Predict. Change one thing. Explain.
Compare the matching and jump cases. Read the two approached heights independently before deciding whether the combined limit exists.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Left limit=2; right limit=4; f(1)=6. Side limits differ, so the two-sided limit does not exist.
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. For x<1 use x+1; for x>1 use the selected branch; f(1)=6. Curves are split at the excluded endpoints, never joined across the break.
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 questionA graph approaches height −1 from the left of x=2 and height 3 from the right, with f(2)=0. State each quantity and justify the two-sided result.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Left limit is −1.
- 1 point: Right limit is 3.
- 1 point: f(2)=0.
- 1 point: The unequal side limits make the two-sided limit nonexistent.
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 does a⁺ mean?
Inputs larger than a approaching a.
RECALL 2Do equal side limits require f(a) to exist?
No.
RECALL 3Is the average of two side limits a two-sided limit?
Only if they already agree; averaging does not repair a jump.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you read a graph from each side?
- A finite two-sided limit exists exactly when both finite one-sided limits exist and agree.
Remember: The superscript minus in 1⁻ means inputs below 1, not negative outputs.
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. For x<1 use x+1; for x>1 use the selected branch; f(1)=6. Curves are split at the excluded endpoints, never joined across the break.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-1.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.3, 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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