Unit 02 · Lesson 02 · Week 4 · Limits: Numerical, Graphical & Algebraic Reasoning
One-Sided Limits and Two-Sided Agreement
A limit describes the output behavior produced by inputs approaching a target. The value at the target may agree with the limit, differ from it, or be undefined. Numerical, graphical, and algebraic evidence should be reconciled. This lesson emphasizes one-sided limits and two-sided agreement through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in one-sided limits and two-sided agreement.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How can values near an input reveal behavior at that input—even when direct substitution fails?
Visual model
A limit describes the output behavior produced by inputs approaching a target. The value at the target may agree with the limit, differ from it, or be undefined. Numerical, graphical, and algebraic evidence should be reconciled.
\lim_{x\to a}f(x)=L\lim_{x\to a^-}f(x)\lim_{x\to a^+}f(x)
Worked reasoning
Misconception check: A limit is not found by merely substituting the target. Substitution works only when the function's local structure permits it.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
One-Sided Limits and Two-Sided Agreement
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Evaluate lim x→3 of (x²−9)/(x−3).
Answer: 6
Reasoning: Factor as (x−3)(x+3); the nearby expression is x+3.
Reasoning: Factor as (x−3)(x+3); the nearby expression is x+3.
A table approaches 4.999 from the left and 5.001 from the right. State the two-sided limit.
Answer: 5
Reasoning: Both one-sided behaviors agree at 5.
Reasoning: Both one-sided behaviors agree at 5.
If the left-hand limit is 2 and the right-hand limit is 7, does the two-sided limit exist?
Answer: No
Reasoning: The one-sided limits must agree.
Reasoning: The one-sided limits must agree.
Evaluate lim x→0 of sin x/x.
Answer: 1
Reasoning: This fundamental trigonometric limit equals 1 in radians.
Reasoning: This fundamental trigonometric limit equals 1 in radians.
Evaluate lim x→3 of (x+1).
Answer: 4
Reasoning: The linear function is continuous, so direct substitution applies.
Reasoning: The linear function is continuous, so direct substitution applies.
Describe what lim x→a f(x)=L claims.
Answer: Outputs approach L as inputs approach a.
Reasoning: The statement concerns nearby behavior, not necessarily f(a).
Reasoning: The statement concerns nearby behavior, not necessarily f(a).