Calculus I · Limits and Continuity · hub
Limits and Continuity Course Overview
Study limits and continuity from the beginning with a sequential Calculus I textbook, guided examples, quizzes, reviews, and practice exams.
Core textbook
The complete textbook path
Follow 47 core pages in order, from the first neighborhood idea to formal epsilon-delta reasoning. Each chapter mixes explanation, guided examples, short checks, and deliberate review.
Start here: orientation
Meet the purpose, prerequisite skills, and notation of the unit before solving a limit.
Reading lens: What information does limit notation give you, and what does it deliberately leave open?
Chapter 1: What a limit means
Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.
Reading lens: What are nearby outputs doing as the input approaches the target from both sides?
Chapter 2: Finite limits and algebra
Turn indeterminate forms into solvable expressions using substitution, limit laws, factoring, conjugates, and piecewise reasoning.
Reading lens: What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior?
- 11Direct Substitution for LimitsLesson
- 12Limit Laws and How to Use ThemLesson
- 13What 0/0 Means in a LimitLesson
- 14Evaluating Limits by FactoringLesson
- 15Evaluating Radical Limits With ConjugatesLesson
- 16Limits With Complex FractionsLesson
- 17Limits With Absolute Values and Piecewise FunctionsLesson
- 18Finite Limit Decision TreeReference
Chapter 3: Trigonometric limits
Use squeezing, the fundamental sine limit, identities, and scaling to make trigonometric behavior predictable.
Reading lens: Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?
Chapter 4: Infinite behavior
Read vertical, horizontal, and slant asymptotes through sign analysis and dominant-term reasoning.
Reading lens: Is the function growing without bound near a finite input, or settling into end behavior as the input grows?
- 26Infinite Limits ExplainedLesson
- 27Vertical Asymptotes and One-Sided Sign AnalysisLesson
- 28Limits at Infinity and Horizontal AsymptotesLesson
- 29Polynomial and Slant AsymptotesLesson
- 30Radical Limits at InfinityLesson
- 31Infinity Minus Infinity LimitsLesson
- 32Complete Rational Function Asymptote AnalysisLesson
Chapter 5: Continuity
Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.
Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?
- 33Continuity at a PointLesson
- 34Continuity on Intervals and at EndpointsLesson
- 35Which Functions Are Continuous?Reference
- 36Types of DiscontinuityLesson
- 37How to Repair a Removable DiscontinuityLesson
- 38Choosing Parameters for Piecewise ContinuityLesson
- 39Two-Parameter Continuity ProblemsLesson
- 40Intermediate Value TheoremLesson
- 41Bisection Method After the IVTLesson
Chapter 6: Formal limits
Translate the intuitive neighborhood picture into epsilon-delta language, constructive proofs, graph windows, and counterexamples.
Reading lens: How small must the input window be to force every allowed output into the requested tolerance band?
Chapter 7: Synthesis
Bring the unit together by choosing methods, explaining decisions, and correcting weak spots before an exam.
Reading lens: Can you diagnose the limit type and justify a method before beginning the algebra?
Practice around the path
Reviews, quizzes, references, and exams
Use these between chapters, after a confusing lesson, or as a mixed rehearsal before an exam. They support the sequence without interrupting it.
What This Book Is For
This is not a packet of lecture notes. It is meant to be the thing you read when the lecture moved too quickly, the textbook skipped three algebra steps, the homework answer says only "6," and every explanation online assumes you already understand the part you are trying to learn.
A student who can do the prerequisite algebra in the diagnostic should be able to use this unit to learn the first major topic of Calculus I from beginning to end. The unit is designed to support three levels of understanding at the same time:
• Plain-language understanding. What the idea means before the notation begins. • Homework competence. A repeatable method for solving the kinds of problems assigned in a first calculus course. • Exam readiness. Enough conceptual and algebraic control to handle unfamiliar versions of familiar problems without copying a pattern blindly.
Every major concept therefore appears in four forms:
• a plain-language explanation that assumes no prior calculus vocabulary; • a formal mathematical statement; • a fully worked example with no important algebra steps omitted; • practice organized from warm-up to exam level.
Think of this book as a patient tutor who never gets annoyed when you ask, "Why did that denominator disappear?" The answer should be visible on the page. If a solution jumps from line one to line five, it has failed at its job, regardless of how elegant the author felt while writing it.
How to Study With This Unit
Do not read mathematics the way you read a news article. A worked solution only becomes useful when you attempt the next move before seeing it.
For each worked example:
• Cover the solution. • Write down what direct substitution gives. • Name the obstacle: ordinary value, , division by zero, infinity over infinity, a jump, or something else. • Choose a method before uncovering the next line. • Compare your work with the solution and identify the first point where your reasoning differs.
For each exercise set, use three passes:
• Pass 1: Learn. Use the chapter and check answers after small groups of problems. • Pass 2: Remember. Return later and work without notes. • Pass 3: Perform. Work a mixed set under a time limit, with no labels telling you which method to use.
Looking at a solution and thinking "that makes sense" is not the same as being able to produce the solution. Recognition feels like knowledge because the page is doing half the thinking for you. Practice removes the page.
Suggested Pacing
The unit can be used as a two-week intensive review or a four-week first course unit.
| Session | Main work | Minimum practice |
|---|---|---|
| 1 | Average change, instantaneous change, basic limit language | Chapter 1 warm-ups 1--10 |
| 2 | Tables, graphs, function value versus limit | Chapter 1 core 11--24 |
| 3 | One-sided limits and failure modes | Chapter 1 exam practice 25--40 |
| 4 | Direct substitution and limit laws | Chapter 2 warm-ups 1--16 |
| 5 | Factoring and cancellation | Chapter 2 core 17--34 |
| 6 | Radicals, complex fractions, absolute values | Chapter 2 core 35--54 |
| 7 | Squeeze Theorem and trig limits | Chapter 3 exercises 1--38 |
| 8 | Infinite limits and vertical asymptotes | Chapter 4 exercises 1--26 |
| 9 | Limits at infinity and radicals | Chapter 4 exercises 27--50 |
| 10 | Continuity and discontinuity | Chapter 5 exercises 1--26 |
| 11 | Parameter problems and IVT | Chapter 5 exercises 27--52 |
| 12 | Formal definition or optional honors track | Chapter 6 selected exercises |
| 13 | Cumulative review | Chapter 7 review set |
| 14 | Practice exam and correction | Practice Exam A or B |
Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.