[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571645-105":53,"doc-detail-571645-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","harolds-calculus-cheat-sheet-introductory-calculus-limits-continuity-derivatives","Harold’s Calculus Cheat Sheet - Introductory Calculus Limits, Continuity, Derivatives","","Harold’s Calculus Cheat Sheet provides a compact reference for core introductory calculus concepts, centered on limits and continuity, then extending to derivative notation and common differentiation rules. It explains the limit definition with epsilon–delta reasoning and the two-sided criterion for existence, outlines continuity conditions and how to structure proofs, and summarizes key limit properties. It also covers derivative notation (Leibniz, Lagrange, Newton, Euler) and lists frequent derivative formulas, including power, exponential, logarithmic, and trigonometric rules, with additional tips for direct substitution and factoring strategies.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-calculus-cheat-sheet-introductory-calculus-limits-continuity-derivatives/571645/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/harolds-calculus-cheat-sheet-introductory-calculus-limits-continuity-derivatives/571645.png","ImageObject",442,249,{"name":88,"@type":89},"Alexandar","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-26","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How does the cheat sheet define a limit using epsilon and delta?","Question",{"text":109,"@type":110},"It states that lim f(x) = L as x approaches c means: for every ε > 0 there exists a δ > 0 such that if |x − c| \u003C δ, then |f(x) − L| \u003C ε.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What condition determines whether a limit exists?",{"text":114,"@type":110},"A limit exists if the value L obtained from approaching c from the left matches the value from approaching c from the right.",{"name":116,"@type":107,"acceptedAnswer":117},"Which derivative notations are included and what do they represent?",{"text":118,"@type":110},"The sheet lists major notations including Leibniz (dy/dx style), Lagrange (prime notation), Newton dot-style, and Euler notation, alongside formulas for first and second derivatives.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571645,1792083992,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":30,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":101},962090768658,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Harold’s Calculus Cheat Sheet  \n21 August 2026  \n\n| Limits & Continuity |  |  |  |\n| --- | --- | --- | --- |\n| Definition of Limit\u003Cbr>Let f be a function defined on an open interval containing 􀜿 and let 􀜮 be a real\u003Cbr>number. The statement: lim 􀝂(􀝔) = 􀜮􀯫→􀯖\u003Cbr>means that for each 􀟳 > 0 there exists a 􀟜 > 0 such that\u003Cbr>if |􀝔 − 􀜿| \u003C 􀟜, then |􀝂(􀝔) − 􀜮| \u003C 􀟳\u003Cbr>Tip:\u003Cbr>Direct substitution: Plug in 􀝂(􀜿) and see if it provides a legal answer. If so, then 􀜮 = 􀝂 (􀜿). | |  |  |\n| The Existence of a Limit\u003Cbr>The limit of 􀝂(􀝔) as 􀝔 approaches c is L if and only if the limit is the same L from both directions. | lim 􀝂(􀝔)􀯫→􀯖 | ⇒ lim 􀯫→􀯖 − | 􀝂 (􀝔) = lim 􀝂 (􀝔) = 􀜮\u003Cbr>􀯫→􀯖+ |\n|  | (Bonus if lim 􀝂(􀝔) = 􀝂 (􀜿) = 􀜮 , but not required.)􀯫 =􀯖 |  |  |\n| Continuous Functions\u003Cbr>If 􀝂(􀝔) is continuous at 􀜿 then lim 􀝂(􀜿) =\u003Cbr>􀯫→􀯖\u003Cbr>􀝂 (􀜿) . | Prove that 􀢌(􀢞) = 􀢞􀫛 − 􀫚 is a continuous function.\u003Cbr>|􀝂(􀝔) − 􀝂(􀜿)| \u003C 􀟳\u003Cbr>= | (􀝔2 − 1) − (􀜿2 − 1)| \u003C 􀟳\u003Cbr>= |􀝔2 − 1 − 􀜿 2 + 1| \u003C 􀟳\u003Cbr>= |􀝔2 − 􀜿2 | \u003C 􀟳\u003Cbr>= | (􀝔 + 􀜿) ∙ (􀝔 − 􀜿)| \u003C 􀟳\u003Cbr>= | (􀝔 + 􀜿)| ∙ | (􀝔 − 􀜿)| \u003C 􀟳\u003Cbr>= |􀝔 + 􀜿| ∙ |􀝔 − 􀜿| \u003C 􀟳\u003Cbr>Since |􀝔 + 􀜿| ≤ |2􀜿| (worst-case scenario)\u003Cbr>= |2􀜿| ∙ |􀝔 − 􀜿| \u003C 􀟝\u003Cbr>= |2􀜿| 􀟜 \u003C 􀟝\u003Cbr>So, given 􀟝 > 0, we can choose 􀢾 = |􀫛􀫚􀢉 | 􀢿 > 􀫙 in the Definition of Continuity. So, substituting the chosen 􀟜 for |􀝔 − 􀜿| we get:\u003Cbr>|􀝂(􀝔) − 􀝂(􀜿)| ≤ |2􀜿| (|12􀜿 | 􀟝 ) = 􀟝 Since both conditions are met, 􀝂(􀝔) is continuous. |  |  |\n| Definition of Continuity\u003Cbr>A function 􀢌 is continuous at 􀜿 if for every 􀟝 > 0 there exists a 􀟜 > 0 such that\u003Cbr>|􀝔 − 􀜿| \u003C 􀟜 and |􀝂 (􀝔) − 􀝂(􀜿)| \u003C 􀟝 .\u003Cbr>Tip:\u003Cbr>Rearrange |􀝂(􀝔) − 􀝂(􀜿)| to have | (􀝔 − 􀜿)| asa factor. Since |􀝔 − 􀜿| \u003C 􀟜 we can find an equation that relates both 􀟜 and 􀟝 together. |  |  |  |\n| Special Trig Limit | lim 􀝏􀝅􀝊 􀝔 = 1 􀯫→0 􀝔 |  |  |\n\n\n| Limit Properties |  |\n| --- | --- |\n| 1. Constant | lim 􀜿 = 􀜿􀯫→􀯔 |\n| 2. Identity Property | lim 􀝔 = 􀜽􀯫→􀯔 |\n| 3. Constant Multiple | lim [􀜿 􀝂(􀝔)] = 􀜿 lim 􀝂(􀝔)\u003Cbr>􀯫→􀯔 􀯫→􀯔 |\n| 4. Sum and Difference | lim [􀝂(􀝔) ± 􀝃 (􀝔)] = lim 􀝂(􀝔) ± lim 􀝃 (􀝔)\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |\n| 5. Product | lim [􀝂(􀝔) ∙ 􀝃 (􀝔)] = lim 􀝂(􀝔) ∙ lim 􀝃 (􀝔)\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |\n| 6. Quotient | l􀯫i [􀝂􀝃((􀝔􀝔))] = l~~􀯫~~l􀯫~~ ~~􀝂􀝃((􀝔􀝔)) , 􀝌􀝎􀝋􀝒􀝅􀝀􀝁􀝀 l􀯫i 􀝃 (􀝔) ≠ 0 |\n| 7. Exponent – Constant | l􀯫i [􀝂(􀝔)]􀯡 = [l􀯫i 􀝂 (􀝔)]􀯡 |\n| 8. Exponent – Function | l􀯫i [􀝂 (􀝔)]􀯚 (􀯫) = [l􀯫i 􀝂 (􀝔)]l􀳣 􀯚 (􀯫) |\n| 9. nth Root | \u003Cbr>l􀯫i 􀳙√􀝂(􀝔) = 􀳙√l􀯫i 􀝂 (􀝔) |\n| 10. Composite Function | lim (􀝂 ∘ 􀝃)(􀝔) = lim 􀝂(􀝃 (􀝔)) = 􀝂 (lim 􀝃 (􀝔))\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |\n\n\n| Derivative Notation |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| First Derivative of a Function\u003Cbr>(Slope Function / Difference Quotient) | 􀢌′(􀢞) = 􀜔􀜑􀜕 􀢌 (􀢞 + 􀢎) − 􀢌(􀢞)\u003Cbr>􀢎→􀫙 􀢎\u003Cbr>􀝂 ′(􀝔) = lim 􀝂 (􀝔 + ℎ) − 􀝂(􀝔 − ℎ)\u003Cbr>ℎ→0 2ℎ\u003Cbr>􀝂 ′(􀜿) = lim 􀝂 (􀝔) − 􀝂(􀜿)\u003Cbr>􀯫→􀯖 􀝔 − 􀜿 |  | |  |  |  |  |\n| Second Symmetric Derivative of a Function\u003Cbr>(Concavity Function) | 􀝂 ′′(􀝔) = lim 􀝂 ′(􀝔 + ℎ) − 􀝂′(􀝔)\u003Cbr>ℎ→0 ℎ\u003Cbr>􀝂 ′′(􀝔)\u003Cbr>􀝂 (􀝔 + ℎ) − 2􀝂(􀝔) + 􀝂(􀝔 − ℎ)\u003Cbr>= lℎi ℎ 2 |  | |  |  |  |  |\n| Notation | Leibniz | Lagrange |  | Newton |  |  | Euler |\n| 1st Derivative Notation | 􀝂 ′(􀝔) = 􀝀􀝀􀝔 􀝂 (􀝔) | 􀝕 ′ |  | 􀝔̇ = |  | 􀝀􀝔\u003Cbr>􀝀􀝐 | 􀜦􀯫 [􀝕] |\n| 2nd Derivative Notation | 􀝂 ′′(􀝔) = 􀝀􀝀􀝔22 􀝂 (􀝔) | 􀝕 ′′ |  | 􀝔̈ = | 􀝀 2􀝔\u003Cbr>􀝀􀝐 2 |  | 􀜦2􀯫􀝂 (􀝔) |\n| nth Derivative Notation | 􀝂 (􀯡) (􀝔) = 􀝀􀝀􀝔􀯡􀯡 􀝂 (􀝔) | 􀝕 (􀯡) |  | 􀝔⃛ = | 􀝀 3􀝔\u003Cbr>􀝀􀝐 3 |  | 􀜦􀯡􀯫􀝂 (􀝔) |\n\n\n| Common Derivatives | (See Cengage Learning 1-Page Calculus Formulas) |  |  |  |  |\n| --- | --- | --- | --- | --- | --- |\n| 1. Constant Rule | 􀝀􀝀􀝔 [􀜿] = 0 |  |  |  |  |\n| 2. Constant Multiple Rule | 􀝀􀝀􀝔 [􀜿􀝂 (􀝔)] = 􀜿􀝂 ′(􀝔) |  |  |  |  |\n| 3. Sum and Difference Rule | 􀝀􀝀􀝔 [􀝂 ± 􀝃] = 􀝂 ′ ± 􀝃 ′ |  |  |  |  |\n| 4. Product Rule | 􀝀􀝀􀝔 [􀝂􀝃] = 􀝂􀝃 ′ + 􀝃 􀝂 ′ |  |  |  |  |\n| 5. Quotient Rule | 􀝀􀝀􀝔 [􀝂􀝃] = 􀝃􀝂~~ ~~′~~ ~~2~~ ~~􀝂􀝃~~ ~~′ |  |  |  |  |\n| 6. Chain Rule | 􀝀􀝀􀝔 [(􀝂 ∘ 􀝃)(􀝔)] = 􀝀􀝀􀝔 [􀝂 (􀝃 (􀝔))] = 􀝂 ′(􀝃 (􀝔)) ∙ 􀝃 ′(􀝔)􀝀􀝕 􀝀􀝕 􀝀􀝐\u003Cbr>~~ ~~ = ~~ ~~ · ~~ ~~\u003Cbr>􀝀􀝔 􀝀􀝐 􀝀􀝔 |  |  |  |  |\n| 7. Power Rule\u003Cbr>􀜲􀯡 (􀝔) = 􀜽􀯡 􀝔 􀯡 + 􀜽􀯡−1􀝔 􀯡−1 + ⋯ + 􀜽1􀝔 + 􀜽0 | 􀢊􀢊􀢞 [􀢞􀢔 ] = 􀢔􀢞􀢔−􀫚 |  |  |  | 􀝀􀝀􀝔 [􀜿􀝔􀯡 ] = 􀝊􀜿􀝔􀯡−1 |\n|  | (Can be used for roots if n is ","cbCaijnD4tOmzaT6","https://ap.wps.com/l/cbCaijnD4tOmzaT6","pdf",1385739,"English","# Limits & Continuity\n## Definition of Limit\n## The Existence of a Limit\n## Continuous Functions\n## Definition of Continuity\n## Special Trig Limit\n## Limit Properties\n# Derivatives\n## Derivative Notation\n## Common Derivatives","[{\"question\":\"How does the cheat sheet define a limit using epsilon and delta?\",\"answer\":\"It states that lim f(x) = L as x approaches c means: for every ε \\u003e 0 there exists a δ \\u003e 0 such that if |x − c| \\u003c δ, then |f(x) − L| \\u003c ε.\"},{\"question\":\"What condition determines whether a limit exists?\",\"answer\":\"A limit exists if the value L obtained from approaching c from the left matches the value from approaching c from the right.\"},{\"question\":\"Which derivative notations are included and what do they represent?\",\"answer\":\"The sheet lists major notations including Leibniz (dy/dx style), Lagrange (prime notation), Newton dot-style, and Euler notation, alongside formulas for first and second derivatives.\"}]","Harold’s Calculus Cheat Sheet - Introductory Calculus Limits, Continuity, Derivatives | PDF",1791922908]