When can you combine limits algebraically?
You will be able to: Apply sum, product and quotient laws with their conditions.
When can you combine limits algebraically?
Two sensor outputs approach 4 and 2. Their sum approaches 6, and their ratio approaches 2 because the denominator’s limiting value stays away from zero.
A useful starting point: What can a table tell you about a limit? →
Words and symbols before equations
- Limit law
- A justified rule for combining existing limits.
- Finite limit
- A real-number limiting value.
- Quotient
- One expression divided by another.
- Nonzero denominator limit
- The condition required for the quotient law.
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. Supplied finite component limits A and B, not inferred from a sample. Quotient law is reported as inapplicable when B=0.
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.
- All three laws apply to these supplied finite limits; the denominator limit is nonzero.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the chemistry
If f→A and g→B are finite, f+g→A+B, f−g→A−B and fg→AB. Fixed constants can be factored through a limit.
For f/g, the quotient law also needs B≠0. If B=0, the law does not apply; the result might be a finite number, unbounded or nonexistent.
Polynomial limits can be found by substitution because powers, sums and constant multiples preserve limits. Rational substitution is valid where the limiting denominator is nonzero.
The laws require the stated component limits. Do not split a difference of two unbounded terms as though infinity were a real number.
A worked example, step by step
Given f→4 and g→−2, find the limit of (3f+g)/g.
- Both supplied component limits are finite.
- The numerator approaches 3(4)+(−2)=10.
- The denominator approaches −2, which is nonzero.
- The quotient limit is 10/(−2)=−5.
A denominator approaching zero is a reason to investigate, not permission to divide by zero.
If B=0, must f/g have no limit?
Compare with an explanation
No. The quotient law is inconclusive; for example x/x→1 as x→0 even though both component limits are zero.
Predict. Change one thing. Explain.
Change the supplied limit B, including zero. Compare the sum and product readouts with the quotient’s explicit condition failure.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
All three laws apply to these supplied finite limits; the denominator limit is nonzero.
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. Supplied finite component limits A and B, not inferred from a sample. Quotient law is reported as inapplicable when B=0.
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 questionIf f→5 and g→−1, evaluate 2f−3g and f/g. State the extra condition for the second calculation.
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Compare with the answer and four-point rubric
- 1 point: 2f−3g approaches 10+3.
- 1 point: Its limit is 13.
- 1 point: f/g approaches −5.
- 1 point: The denominator limit −1 is nonzero, so the quotient law applies.
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 1Can constants pass through limits?
Yes, for the usual finite-limit laws.
RECALL 2What condition belongs to the quotient law?
The denominator limit is nonzero.
RECALL 3Is infinity a real limit usable in ordinary arithmetic laws?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When can you combine limits algebraically?
- For finite A and B: lim(f+g)=A+B and lim(fg)=AB.
- lim(f/g)=A/B requires B≠0.
Remember: A denominator approaching zero is a reason to investigate, not permission to divide by zero.
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. Supplied finite component limits A and B, not inferred from a sample. Quotient law is reported as inapplicable when B=0.
Refresh Kid · AP Calculus AB Unit 1 · Objectives LIM-1.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.5, LIM-1.D. 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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