--- title: Limits description: Limits describe the behavior of a function as its input approaches a specific value. You will understand how to evaluate limits algebraically and graphically, identify continuity, and establish the foundational concepts necessary for defining derivatives and integrals. category: mathematics subcategory: calculus difficulty: beginner, intermediate, advanced url: /subject/limits --- # Limits Limits describe the behavior of a function as its input approaches a specific value. You will understand how to evaluate limits algebraically and graphically, identify continuity, and establish the foundational concepts necessary for defining derivatives and integrals. ## Available Resources 5 Courses • 7 Websites ## Websites ### 1. Jiří Lebl's Real Analysis Book (Free PDF) Open-source undergraduate analysis textbook, free as PDF and used in university courses. Builds sequences, limits and continuity from the completeness of the reals with full proofs and exercises, so a reader can work epsilon-delta arguments unaided. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.jirka.org/ra/realanal.pdf **Tags:** real-analysis, limits, sequences, proofs, free-textbook ### 2. ProofWiki: Real Analysis Category page on ProofWiki indexing definitions and worked proofs across real analysis, including limits of sequences and functions. Each statement links to its full proof and dependencies, useful for checking an argument you half-remember. **Difficulty:** Beginner | **Price:** Free **Link:** https://proofwiki.org/wiki/Category:Real_Analysis **Tags:** real-analysis, proofs, limits, reference, mathematics ### 3. Brilliant.org Limits Interactive lessons on Brilliant covering limits, indeterminate forms and continuity through guided problems rather than lectures. Builds intuition for what a limit means before formal epsilon-delta work; most of the platform sits behind a paid subscription. **Difficulty:** Beginner | **Price:** Paid **Link:** https://brilliant.org/courses/limits/ **Tags:** limits, calculus, interactive, problem-solving, continuity ### 4. Khan Academy Limits Khan Academy's calculus unit on limits and continuity: short videos paired with graded practice on one-sided limits, indeterminate forms, squeeze theorem and continuity. Free, and enough to handle standard AP-style limit computations confidently. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/calculus-1/cs1-limits-and-continuity **Tags:** limits, continuity, calculus, practice-problems, free ### 5. Paul's Online Math Notes Free calculus notes from Lamar University covering the definition of a limit, one-sided limits, limit laws, infinite limits and continuity, each with fully worked examples and practice problems so you can evaluate limits by hand confidently. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/classes/calci/limitsintro.aspx **Tags:** limits, calculus, continuity, lecture-notes, worked-examples ### 6. Khan Academy - Calculus Applications Khan Academy's Calculus 1 course covers derivatives, integrals, and their applications to optimization, related rates, and area/volume problems, with practice exercises and video explanations for each topic. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/calculus-1 **Tags:** calculus, derivatives, integrals, optimization ### 7. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Introduction to Real Analysis Learn real analysis fundamentals! This course, by William F. Trench, covers limits and essential concepts. **Difficulty:** Advanced | **Price:** Free **Link:** https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench) **Tags:** real-analysis, limits, continuity, open-textbook, proofs ### 2. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 3. Single Variable Calculus (MIT 18.01SC) **Author:** David Jerison MIT's self-paced 18.01SC covers differentiation, applications of derivatives, integration, techniques of integration, and infinite series, with problem sets, recitation videos, and exams with solutions. Completing it gives the standard first-semester calculus foundation. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ **Tags:** single-variable-calculus, derivatives, integrals, infinite-series ### 4. Calculus 1: Limits and Continuity University of Sydney course, taught by David Easdown, moving from limits and the Mean Value Theorem through integration techniques, sequences and Taylor series, to first and second-order differential equations, with roughly forty hours of graded practice. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/introduction-to-advanced-calculus **Tags:** courses, mathematics-statistics, calculus ### 5. Introduction to Complex Analysis **Author:** Petra Bonfert-Taylor Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/complex-analysis **Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/limits