[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571646-105":53,"doc-detail-571646-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","calculus-cheat-sheet-limits-and-derivatives","Calculus Cheat Sheet - Limits and Derivatives","","Calculus Cheat Sheet consolidates key results and evaluation methods for limits, one-sided limits, continuity, the intermediate value theorem, and core derivative concepts. It presents precise and working definitions of limits, including right-hand, left-hand, and infinite limits, then summarizes limit relationships and properties. It also lists fundamental limit evaluations, techniques for continuous functions and composition, algebraic simplification strategies, L’Hospital’s rule, and derivative definitions, notation, interpretation, and basic differentiation formulas.",{"@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/calculus-cheat-sheet-limits-and-derivatives/571646/",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/calculus-cheat-sheet-limits-and-derivatives/571646.png","ImageObject",442,249,{"name":88,"@type":89},"Damian","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","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},"What is the precise definition of a limit, and how does the “working” definition differ?","Question",{"text":109,"@type":110},"The precise definition states that f(x) approaches L as x approaches a using an ε-δ condition. The working definition expresses the same idea as making f(x) arbitrarily close to L by choosing x sufficiently close to a (without taking x = a).","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are right-hand and left-hand limits defined?",{"text":114,"@type":110},"A right-hand limit lim+ f(x)=L uses the same limit requirement but restricts x to be greater than a. A left-hand limit lim- f(x)=L restricts x to be less than a.",{"name":116,"@type":107,"acceptedAnswer":117},"How is the derivative defined and what does it represent?",{"text":118,"@type":110},"If y=f(x), the derivative is defined as f'(x)=lim(h→0) [f(x+h)-f(x)]/h. It represents the slope of the tangent line, the instantaneous rate of change, and—if f(x) is position—the velocity at time x=a.","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},571646,1792009231,{"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":8,"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":79},137451208677,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Calculus Cheat Sheet  \nLimits  \nDefinitions  \nPrecise Definition : We say limf ( x ) = L if  \nxÆa  \nfor every e > 0 there is a d > 0 such that whenever 0 \u003C x-a  \u003C d then f ( x )-L \u003C e .  \n“Working” Definition : We say limf ( x ) = L  \nxÆa  \nif we can make f ( x ) as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x = a .  \nRight hand limit : lim+ f ( x ) = L . This has xÆa  \nthe same definition as the limit except it requires x > a .  \nLeft hand limit : lim-f ( x ) = L . This has the xÆa  \nsame definition as the limit except it requires x \u003C a .  \nLimit at Infinity : We say lim f ( x ) = L if we  \nxÆ•  \ncan make f ( x ) as close to L as we want by taking x large enough and positive.  \nThere is a similar definition for lim f ( x ) = L  \nxÆ-•  \nexcept we require x large and negative. Infinite Limit : We say limf ( x ) =• if we  \nxÆa  \ncan make f ( x ) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x = a .  \nThere is a similar definition for lim f ( x ) = -•  \nxÆa  \nexcept we make f ( x ) arbitrarily large and negative.  \nRelationship between the limit and one-sided limits  \nlimf ( x ) = L fi lim f ( x ) = lim f ( x ) = L lim f ( x ) = lim f ( x ) = L fi limf ( x ) = L  \nxÆa xÆa+ xÆa- xÆa+ xÆa- xÆa  \nlim+ f ( x ) 􀂹 lim-f ( x ) fi lim f ( x ) Does Not Exist  \nxÆa xÆa xÆa  \nProperties  \nAssume limf ( x ) and limg ( x ) both exist and c is any number then,  \nxÆa xÆa  \n1.  cf ( x ) = c  f ( x )  \n2.  f ( x ) ± g ( x ) = f ( x ) ±  g ( x )  \n3.  f ( x ) g ( x ) =  f ( x ) g ( x )  \n4.  ÈÍÎ~~ ~~fg~~ (~~(~~ ~~xx~~)~~)~~ ~~ ˘˚˙ = ~~ ~~aa~~ ~~fg~~ (~~(xx~~)~~) provided 5.  f ( x )n =  f ( x )n  \n6.  ( x )  = n f ( x )  \nlim g ( x ) 􀂹 0  \nxÆa  \nBasic Limit Evaluations at ± •  \nNote : sgn ( a ) = 1 if a > 0 and sgn ( a ) = -1 if a \u003C 0 .  \n1. lim ex =• & lim ex = 0 xÆ• xÆ -•  \n2. lim ln ( x ) =• & lim ln ( x ) = -• xÆ• xÆ0 +  \n3. If r > 0 then lim ~~ ~~~~ ~~ = 0 xÆ• x  \n4. If r > 0 and xr is real for negative x then lim ~~ ~~~~ ~~ = 0  \nxÆ-• x  \n5. n even : lim xn =•  \nxƱ•  \n6. n odd : lim xn =• & lim xn = -• xÆ• xÆ -•  \n7. n even : x• a xn + L + bx + c = sgn ( a )•  \n8. n odd : lim a xn + L + bx + c = sgn ( a )• xÆ•  \n9. n odd : lim a xn + L + c x + d = -sgn ( a )• xÆ-•  \nVisit [http://tutorial.math.lamar.edu](http://tutorial.math.lamar.edu) for a complete set of Calculus notes. © 2005 Paul Dawkins  \nCalculus Cheat Sheet  \nEvaluation Techniques  \nContinuous Functions  \nIff ( x ) is continuous at a then lim f ( x ) = f ( a )  \nxÆa  \nContinuous Functions and Composition  \nf ( x ) is continuous at b and lim g ( x ) = b then  \nxÆa  \n f ( g ( x )) = f (  g ( x ) ) = f ( b )  \nFactor and Cancel  \n2 ~~ ~~x2~~ ~~+x24-x212 = 2 ~~ ~~(~~ ~~x-x)~~ ~~-(~~ ~~x2+~~) ~~6~~ ~~)  \n= lim  x + 6 =  8 = 4  \nxÆ2 x 2  \nRationalize Numerator/Denominator  \n9 ~~ ~~3x2--x81 = 9 ~~ ~~3x2--x81  \n3 +    \n3 +   \n= 9 ~~ ( ~~x~~ ~~2-891~~)~~-~~(~~x3+~~ ~~x~~  )~~ = 9 ~~ ( ~~x~~ ~~+~~ ~~9~~)~~-~~( ~~13+~~ ~~x~~  )~~  \n-1 1 = ( 18) ( 6 ) = -108  \nCombine Rational Expressions  \n0 ~~ ~~1h~~ ~~  ~~ ~~x~~ ~~~~ ~~h- ~~ ~~1xˆ¯˜ = 0 ~~ ~~1h ÁËÊ ~~ ~~xx-~~( ~~h~~) ~~)~~ ~~ ˆ¯˜     \n1 Ê -h ˆ˜ -1 1 = 0 h ËÁ x ( x + h) ¯˜ = 0 x ( x + h) = -x 2  \nL’Hospital’s Rule  \nIf  ~~ ~~fg~~ (~~(xx~~)~~) = ~~ ~~00 or  ~~ ~~fg~~ (~~(xx~~)~~) = ~~ ~~•• then,  ~~ ~~fg~~ (~~(xx~~)~~) =  ~~ ~~fg¢¢~~(~~(~~ ~~xx~~)~~) a is a number,• or-•  \nPolynomials at Infinity  \np ( x ) and q ( x ) are polynomials. To compute x pq~~(~~(xx~~)~~) factor largest power of x in q ( x ) out of both p ( x ) and q ( x ) then compute limit.  \nxm- 3x5x-2-24x2~~ ~~ = xm- xx22~~ ~~(~~(~~~~ ~~--4x22~~)~~) = m- ~~ ~~--4x22~~  ~~ = - ~~ ~~32  \nPiecewise Function  \n-2 g ( x ) where g ( x ) = ÌÏxÓ1-2 +35x if xif x  --22  \nCompute two one sided limits, lim-g ( x ) = lim-x2 + 5 = 9 xÆ-2 xÆ-2  \nlim+ g ( x ) = lim+ 1-3x = 7 xÆ-2 xÆ-2  \nOne sided limits are different so -2 g ( x ) doesn’t exist. If the two one sided limits had been equa","cbCaibyjAxQZ2P6S","https://ap.wps.com/l/cbCaibyjAxQZ2P6S","pdf",251115,"English","# Limits\n## Definitions\n## Relationship between limits and one-sided limits\n## Properties\n## Basic limit evaluations\n# Evaluation Techniques\n## Continuous functions\n## Factor and cancel\n## Rationalize\n## Combine rational expressions\n## L’Hospital’s Rule\n## Polynomials at infinity\n## Piecewise function\n## Some continuous functions\n## Intermediate Value Theorem\n# Derivatives\n## Definition and notation\n## Interpretation of the derivative\n## Basic properties and formulas","[{\"question\":\"What is the precise definition of a limit, and how does the “working” definition differ?\",\"answer\":\"The precise definition states that f(x) approaches L as x approaches a using an ε-δ condition. The working definition expresses the same idea as making f(x) arbitrarily close to L by choosing x sufficiently close to a (without taking x = a).\"},{\"question\":\"How are right-hand and left-hand limits defined?\",\"answer\":\"A right-hand limit lim+ f(x)=L uses the same limit requirement but restricts x to be greater than a. A left-hand limit lim- f(x)=L restricts x to be less than a.\"},{\"question\":\"How is the derivative defined and what does it represent?\",\"answer\":\"If y=f(x), the derivative is defined as f'(x)=lim(h→0) [f(x+h)-f(x)]/h. It represents the slope of the tangent line, the instantaneous rate of change, and—if f(x) is position—the velocity at time x=a.\"}]","Calculus Cheat Sheet - Limits and Derivatives | PDF",1791922908]