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Calculus course topic

Limits & Continuity

How functions behave near a point, when substitution is legal, and what continuity actually promises.

Core textbook

The complete textbook path

Follow the core path from the first neighborhood idea to formal epsilon-delta reasoning. Each section mixes explanation, guided examples, short checks, and deliberate review.

01–03

Start here: orientation

Meet the purpose, prerequisite skills, and notation of the unit before solving a limit.

  1. 01Limits and Continuity Course OverviewOverview
  2. 02Calculus Limits Prerequisite DiagnosticDiagnostic
  3. 03Limit Notation GuideReference
04–10

Section 1: What a limit means

Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.

  1. 04Why Limits Matter in CalculusLesson
  2. 05What a Limit MeansLesson
  3. 06Limits at Holes and Undefined PointsLesson
  4. 07Function Value vs. LimitLesson
  5. 08One-Sided Limits From the Left and RightLesson
  6. 09When a Limit Does Not ExistLesson
  7. 10How to Read Limits From a GraphLesson
11–18

Section 2: Finite limits and algebra

Turn indeterminate forms into solvable expressions using substitution, limit laws, factoring, conjugates, and piecewise reasoning.

  1. 11Direct Substitution for LimitsLesson
  2. 12Limit Laws and How to Use ThemLesson
  3. 13What 0/0 Means in a LimitLesson
  4. 14Evaluating Limits by FactoringLesson
  5. 15Evaluating Radical Limits With ConjugatesLesson
  6. 16Limits With Complex FractionsLesson
  7. 17Limits With Absolute Values and Piecewise FunctionsLesson
  8. 18Finite Limit Decision TreeReference
19–25

Section 3: Trigonometric limits

Use squeezing, the fundamental sine limit, identities, and scaling to make trigonometric behavior predictable.

  1. 19The Squeeze TheoremLesson
  2. 20Why sin(x)/x Approaches 1Lesson
  3. 21Geometric Proof of the sin(x)/x LimitLesson
  4. 22Scaled Sine and Tangent LimitsLesson
  5. 23Limits of Ratios of Sine FunctionsLesson
  6. 24Cosine Limits and Trigonometric IdentitiesLesson
  7. 25Trigonometric Limit Decision GuideReference
26–32

Section 4: Infinite behavior

Read vertical, horizontal, and slant asymptotes through sign analysis and dominant-term reasoning.

  1. 26Infinite Limits ExplainedLesson
  2. 27Vertical Asymptotes and One-Sided Sign AnalysisLesson
  3. 28Limits at Infinity and Horizontal AsymptotesLesson
  4. 29Polynomial and Slant AsymptotesLesson
  5. 30Radical Limits at InfinityLesson
  6. 31Infinity Minus Infinity LimitsLesson
  7. 32Complete Rational Function Asymptote AnalysisLesson
33–41

Section 5: Continuity

Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.

  1. 33Continuity at a PointLesson
  2. 34Continuity on Intervals and at EndpointsLesson
  3. 35Which Functions Are Continuous?Reference
  4. 36Types of DiscontinuityLesson
  5. 37How to Repair a Removable DiscontinuityLesson
  6. 38Choosing Parameters for Piecewise ContinuityLesson
  7. 39Two-Parameter Continuity ProblemsLesson
  8. 40Intermediate Value TheoremLesson
  9. 41Bisection Method After the IVTLesson
42–46

Section 6: Formal limits

Translate the intuitive neighborhood picture into epsilon-delta language, constructive proofs, graph windows, and counterexamples.

  1. 42Epsilon-Delta Definition: An IntroductionLesson
  2. 43Epsilon-Delta Proofs for Linear FunctionsLesson
  3. 44Epsilon-Delta Proofs for Quadratic FunctionsLesson
  4. 45Reading Epsilon and Delta From a GraphLesson
  5. 46How to Disprove a Claimed LimitLesson
47–47

Section 7: Synthesis

Bring the unit together by choosing methods, explaining decisions, and correcting weak spots before an exam.

  1. 47Limits and Continuity Unit ReviewReview

Practice around the path

Reviews, quizzes, references, and exams

Use these between sections, after a confusing lesson, or as a mixed rehearsal before an exam. They support the sequence without interrupting it.

ReviewLimit Meaning ReviewReview Limit Meaning Review with a concise concept summary, common errors, and links to targeted practice.Concept quizLimit Meaning Concept QuizTest Limit Meaning Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setLimit Meaning Practice ProblemsPractice Limit Meaning Practice Problems with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ReviewFinite Limits ReviewReview Finite Limits Review with a concise concept summary, common errors, and links to targeted practice.Concept quizFinite Limits Concept QuizTest Finite Limits Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setFinite Limit Practice ProblemsPractice Finite Limit Practice Problems with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ReviewSqueeze and Trigonometric Limits ReviewReview Squeeze and Trigonometric Limits Review with a concise concept summary, common errors, and links to targeted practice.Concept quizTrigonometric Limits Concept QuizTest Trigonometric Limits Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setTrigonometric Limit Practice ProblemsPractice Trigonometric Limit Practice Problems with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ReviewInfinite Limits and Asymptotes ReviewReview Infinite Limits and Asymptotes Review with a concise concept summary, common errors, and links to targeted practice.Concept quizInfinite Limits Concept QuizTest Infinite Limits Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setInfinite Limits and Asymptote PracticePractice Infinite Limits and Asymptote Practice with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ReviewContinuity and IVT ReviewReview Continuity and IVT Review with a concise concept summary, common errors, and links to targeted practice.Concept quizContinuity Concept QuizTest Continuity Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setContinuity and IVT Practice ProblemsPractice Continuity and IVT Practice Problems with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ExtensionFormal Definition of an Infinite LimitGo beyond the standard unit with Formal Definition of an Infinite Limit, including formal definitions, worked reasoning, and optional proof practice.ReviewEpsilon-Delta ReviewReview Epsilon-Delta Review with a concise concept summary, common errors, and links to targeted practice.Concept quizEpsilon-Delta Concept QuizTest Epsilon-Delta Concept Quiz with short answer checks, hints, worked feedback, and links back to the exact lessons you need.Practice setEpsilon-Delta Practice ProblemsPractice Epsilon-Delta Practice Problems with warm-up, homework-level, reasoning, and exam-style problems plus answer support.ReviewCommon Limits Exam ErrorsReview Common Limits Exam Errors with a concise concept summary, common errors, and links to targeted practice.Study guideHow to Correct a Calculus Practice TestUse How to Correct a Calculus Practice Test to turn missed calculus questions into a concrete study plan instead of merely rereading solutions.Practice setCumulative Limits and Continuity PracticePractice Cumulative Limits and Continuity Practice with warm-up, homework-level, reasoning, and exam-style problems plus answer support.Practice examLimits and Continuity Practice Exam ATake Limits and Continuity Practice Exam A as a realistic first-unit Calculus I assessment covering limits, asymptotes, continuity, and the IVT.Practice examLimits and Continuity Practice Exam BTake Limits and Continuity Practice Exam B as a realistic first-unit Calculus I assessment covering limits, asymptotes, continuity, and the IVT.

Check your work

Exam answer keys

Finish an honest attempt, then compare one problem at a time. Every supplied exam answer is published and source-traced.

Complete keyPractice Exam A Answer KeyCheck all 18 answers for Limits and Continuity Practice Exam A, with concise reasoning and source-traced solutions.Open answer key →Complete keyPractice Exam B Answer KeyCheck all 14 answers for Limits and Continuity Practice Exam B, with concise reasoning and source-traced solutions.Open answer key →

Further exploration

Deep dives and extra articles

The core map above is the textbook. These articles are side trails: close readings of one method, a single integral, or a conceptual distinction that rewards more time and more examples.

Understand the idea

Direct answers and concept explanations that make the structure visible.

01Calculus · Direct answerWhy is the limit of sin x over x equal to 1?A foundational limit whose real proof comes from geometry, not from plugging in zero and hoping.Calculus IDirect answer03Calculus · Concept explainerWhat continuity at a point really requiresThree conditions, one precise promise: nearby inputs produce nearby outputs without a break at the point.Calculus IConcept explainer05Calculus · Concept explainerInfinite limits and vertical asymptotesInfinity describes unbounded behavior, not a number a function eventually reaches.Calculus IConcept explainer06Calculus · Concept explainerOne-sided limits and jump discontinuitiesA two-sided limit exists only when the function approaches the same value from the left and from the right.Calculus IConcept explainer
Work the method

Repeatable procedures with worked examples and checks.

02Calculus · Method guideHow to evaluate an indeterminate limitA decision process for 0/0 and ∞/∞ that starts with algebra before reaching for a theorem.Calculus IMethod guide04Calculus · Method guideHow and when to use the Squeeze TheoremTrap a difficult function between two easier functions that are forced to meet at the same limit.Calculus IMethod guide