[{"@context":"https:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/blog.terabox.com\/insights\/rational-expressions-difference-quotient-limits#BlogPosting","mainEntityOfPage":"https:\/\/blog.terabox.com\/insights\/rational-expressions-difference-quotient-limits","headline":"Rational Expressions, Difference Quotient & Limits Guide","name":"Rational Expressions, Difference Quotient & Limits Guide","description":"\ud83d\udcfa Today’s recommended deep-dive video: https:\/\/www.youtube.com\/watch?v=HfACrKJ_Y2w Mastering Calculus Foundations: From Rational Algebra to the DerivativeRational Expressions and the Difference QuotientLimits, Continuity, and Infinite BehaviorTrigonometric Mastery and LogarithmsThe Derivative: Rules, Motion, and EconomicsKey TakeawaysQ&A Mastering Calculus Foundations: From Rational Algebra to... ","datePublished":"2026-07-23","dateModified":"2026-07-23","author":{"@type":"Person","@id":"https:\/\/blog.terabox.com\/author\/flextech-admin\/#Person","name":"flextech-admin","url":"https:\/\/blog.terabox.com\/author\/flextech-admin\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/ad516503a11cd5ca435acc9bb6523536?s=150&d=mm&r=gforcedefault=1","url":"https:\/\/secure.gravatar.com\/avatar\/ad516503a11cd5ca435acc9bb6523536?s=150&d=mm&r=gforcedefault=1","height":96,"width":96}},"publisher":{"@type":"Organization","name":"terabox","logo":{"@type":"ImageObject","@id":"http:\/\/blog.terabox.com\/wp-content\/uploads\/2021\/11\/logo\u4ea7\u54c1\u540d-\u7ad6\u7248.png","url":"http:\/\/blog.terabox.com\/wp-content\/uploads\/2021\/11\/logo\u4ea7\u54c1\u540d-\u7ad6\u7248.png","width":900,"height":900}},"image":{"@type":"ImageObject","@id":"https:\/\/img.youtube.com\/vi\/HfACrKJ_Y2w\/maxresdefault.jpg","url":"https:\/\/img.youtube.com\/vi\/HfACrKJ_Y2w\/maxresdefault.jpg","height":"","width":""},"url":"https:\/\/blog.terabox.com\/insights\/rational-expressions-difference-quotient-limits","video":{"@context":"http:\/\/schema.org\/","@type":"VideoObject","@id":"https:\/\/www.youtube.com\/watch?v=HfACrKJ_Y2w#VideoObject","contentUrl":"https:\/\/www.youtube.com\/watch?v=HfACrKJ_Y2w","name":"Calculus 1 - Full College Course","description":"Learn Calculus 1 in this full college course.\n\nThis course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https:\/\/www.youtube.com\/channel\/UCkyLJh6hQS1TlhUZxOMjTFw\n\nThis course combines two courses taught by Dr. Green. She teaches both Calculus 1 and a Calculus 1 Corequisite course, designed to be taken at the same time. The lectures from the Corquisite course, which review important Algebra and Trigonometry concepts, have been interspersed with the Calculus 1 lectures at the places suggested by Dr. Green.\n\n\u2b50\ufe0f Prerequisites \u2b50\ufe0f\n\ud83c\udfa5 Algebra: https:\/\/www.youtube.com\/watch?v=LwCRRUa8yTU\n\ud83c\udfa5 Precalculus: https:\/\/www.youtube.com\/watch?v=eI4an8aSsgw\n\n\u2b50\ufe0f Lecture Notes \u2b50\ufe0f\n\ud83d\udd17 Calculus 1 Corequisite Notes: http:\/\/lindagreen.web.unc.edu\/files\/2020\/08\/courseNotes_math231L_2020Fall.pdf\n\ud83d\udd17 Calculus 1 Notes: http:\/\/lindagreen.web.unc.edu\/files\/2019\/12\/courseNotes_m231_2018_S.pdf\n\n\ud83d\udd17 Course Transcription and Notes (Thanks to @MathTranscribed ): https:\/\/drive.google.com\/file\/d\/1wvmKbC4X-7su7YFbLoUmUebfrh3LXbd1\/view?usp=sharing\n\n\u2b50\ufe0f Course Contents \u2b50\ufe0f\n(0:00:00) [Corequisite] Rational Expressions\n(0:09:40) [Corequisite] Difference Quotient\n(0:18:20) Graphs and Limits\n(0:25:51) When Limits Fail to Exist\n(0:31:28) Limit Laws\n(0:37:07) The Squeeze Theorem\n(0:42:55) Limits using Algebraic Tricks\n(0:56:04) When the Limit of the Denominator is 0\n(1:08:40) [Corequisite] Lines: Graphs and Equations\n(1:17:09) [Corequisite] Rational Functions and Graphs\n(1:30:35) Limits at Infinity and Graphs\n(1:37:31) Limits at Infinity and Algebraic Tricks\n(1:45:34) Continuity at a Point\n(1:53:21) Continuity on Intervals\n(1:59:43) Intermediate Value Theorem\n(2:03:37) [Corequisite] Right Angle Trigonometry\n(2:11:13) [Corequisite] Sine and Cosine of Special Angles\n(2:19:16) [Corequisite] Unit Circle Definition of Sine and Cosine\n(2:24:46) [Corequisite] Properties of Trig Functions\n(2:35:25) [Corequisite] Graphs of Sine and Cosine\n(2:41:57) [Corequisite] Graphs of Sinusoidal Functions\n(2:52:10) [Corequisite] Graphs of Tan, Sec, Cot, Csc\n(3:01:03) [Corequisite] Solving Basic Trig Equations\n(3:08:14) Derivatives and Tangent Lines\n(3:22:55) Computing Derivatives from the Definition\n(3:34:02) Interpreting Derivatives\n(3:42:33) Derivatives as Functions and Graphs of Derivatives\n(3:56:25) Proof that Differentiable Functions are Continuous\n(4:01:09) Power Rule and Other Rules for Derivatives\n(4:07:42) [Corequisite] Trig Identities\n(4:15:14) [Corequisite] Pythagorean Identities\n(4:20:35) [Corequisite] Angle Sum and Difference Formulas\n(4:28:31) [Corequisite] Double Angle Formulas\n(4:36:01) Higher Order Derivatives and Notation\n(4:39:22) Derivative of e^x\n(4:46:52) Proof of the Power Rule and Other Derivative Rules\n(4:56:31) Product Rule and Quotient Rule\n(5:02:09) Proof of Product Rule and Quotient Rule\n(5:10:40) Special Trigonometric Limits\n(5:17:31) [Corequisite] Composition of Functions\n(5:29:54) [Corequisite] Solving Rational Equations\n(5:40:02) Derivatives of Trig Functions\n(5:46:23) Proof of Trigonometric Limits and Derivatives\n(5:54:38) Rectilinear Motion\n(6:11:41) Marginal Cost\n(6:16:51) [Corequisite] Logarithms: Introduction\n(6:25:32) [Corequisite] Log Functions and Their Graphs\n(6:36:17) [Corequisite] Combining Logs and Exponents\n(6:40:55) [Corequisite] Log Rules\n(6:49:27) The Chain Rule\n(6:58:44) More Chain Rule Examples and Justification\n(7:07:43) Justification of the Chain Rule\n(7:10:00) Implicit Differentiation\n(7:20:28) Derivatives of Exponential Functions\n(7:25:38) Derivatives of Log Functions\n(7:29:38) Logarithmic Differentiation\n(7:37:08) [Corequisite] Inverse Functions\n(7:51:22) Inverse Trig Functions\n(8:00:56) Derivatives of Inverse Trigonometric Functions\n(8:12:11) Related Rates - Distances\n(8:17:55) Related Rates - Volume and Flow\n(8:22:21) Related Rates - Angle and Rotation\n(8:28:20) [Corequisite] Solving Right Triangles\n(8:34:54) Maximums and Minimums\n(8:46:18) First Derivative Test and Second Derivative Test\n(8:51:37) Extreme Value Examples\n(9:01:33) Mean Value Theorem\n(9:09:09) Proof of Mean Value Theorem\n(9:14:59) Polynomial and Rational Inequalities\n(9:25:20) Derivatives and the Shape of the Graph\n(9:33:31) Linear Approximation\n(9:48:28) The Differential\n(9:59:11) L'Hospital's Rule\n(10:06:27) L'Hospital's Rule on Other Indeterminate Forms\n(10:16:13) Newtons Method\n(10:27:45) Antiderivatives\n(10:33:24) Finding Antiderivatives Using Initial Conditions\n(10:41:59) Any Two Antiderivatives Differ by a Constant\n(10:45:19) Summation Notation\n(10:49:12) Approximating Area\n(11:04:22) The Fundamental Theorem of Calculus, Part 1\n(11:15:02) The Fundamental Theorem of Calculus, Part 2\n(11:22:17) Proof of the Fundamental Theorem of Calculus\n(11:29:18) The Substitution Method\n(11:38:07) Why U-Substitution Works\n(11:40:23) Average Value of a Function\n(11:47:57) Proof of the Mean Value Theorem\n\nCorrection:\n9:09:10 This section should be this: https:\/\/youtu.be\/L2W-CyRYBB0","thumbnailUrl":["https:\/\/i.ytimg.com\/vi\/HfACrKJ_Y2w\/default.jpg","https:\/\/i.ytimg.com\/vi\/HfACrKJ_Y2w\/mqdefault.jpg","https:\/\/i.ytimg.com\/vi\/HfACrKJ_Y2w\/hqdefault.jpg","https:\/\/i.ytimg.com\/vi\/HfACrKJ_Y2w\/sddefault.jpg","https:\/\/i.ytimg.com\/vi\/HfACrKJ_Y2w\/maxresdefault.jpg"],"uploadDate":"2020-08-25T15:30:08+00:00","duration":"PT11H53M48S","embedUrl":"https:\/\/www.youtube.com\/embed\/HfACrKJ_Y2w","publisher":{"@type":"Organization","@id":"https:\/\/www.youtube.com\/channel\/UC8butISFwT-Wl7EV0hUK0BQ#Organization","url":"https:\/\/www.youtube.com\/channel\/UC8butISFwT-Wl7EV0hUK0BQ","name":"freeCodeCamp.org","description":"Learn math, programming, and computer science for free. 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