When does direct substitution work inside a function?
You will be able to: Use continuity of an outer function to justify a composite limit.
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.
When does direct substitution work inside a function?
If an inner expression approaches 9, its square root approaches 3 because the square-root graph is continuous near 9. The outer rule matters.
A useful starting point: When can you combine limits algebraically? →
Words and symbols before equations
- Composite function
- An outer rule applied to the output of an inner rule.
- Outer function
- The rule applied last.
- Continuity at L
- The outer output near L agrees with its value at L.
- Domain
- Inputs allowed by the formula.
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. As x→0. The square-root case stays in its domain. The outer step rule is discontinuous at the inner limit zero.
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.
- At x=±0.1, outputs are 2.98329 and 3.01662. Inner limit 9 and continuity of sqrt at 9 give limit 3.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
If g(x)→L and F is continuous at L on the relevant domain, then F(g(x))→F(L). This explains why many substitutions work; it is not a universal permission to substitute into any rule.
For sqrt(2x+5) as x→2, the inside approaches 9 and the square root is continuous there. The limit is 3.
For a discontinuous outer step rule, g(x)→L may send outputs to opposite sides of its jump. Knowing g’s limit alone does not justify the same substitution.
Check domains first. A square root needs a nonnegative radicand; a logarithm needs a positive input. A quotient needs its denominator condition too.
A worked example, step by step
Find the limit of ln(x²+1) as x→0.
- The inner polynomial x²+1 approaches 1.
- The logarithm is defined and continuous at the positive input 1.
- Move the limit through the outer logarithm.
- The result is ln(1)=0.
For composition, check the outer function at the inner limit, not only at a convenient nearby point.
Why can we substitute inside ln(x²+1) near zero?
Compare with an explanation
The inner value tends to 1 and ln is continuous at the positive input 1.
Predict. Change one thing. Explain.
Switch between a continuous square-root rule and a jumping outer rule. Explain why equal inner limiting values are not alone sufficient in the jump case.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x=±0.1, outputs are 2.98329 and 3.01662. Inner limit 9 and continuity of sqrt at 9 give limit 3.
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. As x→0. The square-root case stays in its domain. The outer step rule is discontinuous at the inner limit zero.
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 questionEvaluate the limit of sqrt(x+8) as x→1. Identify the inner limit, outer rule, domain condition and conclusion.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The inner expression approaches 9.
- 1 point: The outer rule is square root.
- 1 point: It is defined and continuous near 9 because these inputs are positive.
- 1 point: The composite limit is 3.
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 1Which rule is outer in sqrt(g(x))?
The square root.
RECALL 2Where must the outer continuity be checked?
At the inner limiting value.
RECALL 3What is the domain of ln u?
u>0.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When does direct substitution work inside a function?
- If g→L and F is continuous at L, then F(g)→F(L), with valid domains.
Remember: For composition, check the outer function at the inner limit, not only at a convenient nearby point.
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. As x→0. The square-root case stays in its domain. The outer step rule is discontinuous at the inner limit zero.
Refresh Kid · AP Calculus BC 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 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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