[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522807-105":59,"doc-detail-522807-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","evaluating-limits-limits-at-removable-discontinuities-worksheet","Evaluating Limits -  Limits at Removable Discontinuities - Worksheet","","Worksheet focused on evaluating limits in algebraic expressions and piecewise-defined functions, including problems that require direct substitution and others involving removable discontinuities. Includes a set of step-by-step limit calculations with given formulas and at least one prompt to use provided graphs to sketch corresponding functions. Extends practice with critical thinking items asking for specific values where limits exist even when a rational function is undefined, and where direct evaluation fails versus succeeds.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/evaluating-limits-limits-at-removable-discontinuities-worksheet/522807/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/evaluating-limits-limits-at-removable-discontinuities-worksheet/522807.png","ImageObject",300,407,{"name":92,"@type":93},"Alex Sinclair","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"When can a limit of a rational function be solved using direct evaluation?","Question",{"text":112,"@type":113},"When the function is defined at the target x-value, direct substitution can be used to compute the limit. The worksheet includes prompts that ask for an example value of a where this applies.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does it mean for a limit to exist even if the function is undefined at a point?",{"text":117,"@type":113},"The limit can still exist when the left-hand and right-hand values approach the same number, even though the function has no defined value at that point. The critical thinking question explicitly targets this situation at x = -1.",{"name":119,"@type":110,"acceptedAnswer":120},"How should the graph be used in limits problems on this worksheet?",{"text":121,"@type":113},"For the problems that mention using the provided graph, sketch the behavior of the function near the indicated x-value(s). This helps determine the approaching y-value(s) when direct evaluation is not straightforward.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522807,1791496978,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52,"preview_page_count":4},1099523882182,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Kuta Software-Infinite Calculus Name   \nEvaluating Limits Date  Period   \nEvaluate each limit.  \n1) lx f (x) , f (x) = {−x52, + 2, xx  22  \nx 2 − 4 2) lim − ~~ ~~  \nx→ −2 x + 2  \nEvaluate each limit. You may use the provided graph to sketch the function.  \n3) lim x→ 3  \nx 2 − 7x + 12  \nx − 3  \n\n| | |  |  |  | f(x) |  |  |  |  |  |  |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | 6 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | 4 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | 2 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| − | 4 | − | 2 |  |  | 2 |  | 4 |  | 6 |  |  | 8 | 1 | 0 | x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | −2 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | −4 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | −6 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  | −8 |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\n4) lim  x + 3  x→ −3 x 2 + 2x − 3  \n\n| | |  |  |  |  |  |  |  |  |  | f(x) |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  |  |  |  | 8 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | 6 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | 4 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | 2 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| − | 10 | − | 8 | − | 6 | − | 4 | − | 2 |  |  | 2 |  | 4 |  | x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | −2 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | −4 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | −6 |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  | −8 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\nEvaluate each limit.  \n5) lx f (x) , f (x) = {x2,+ 1, xx  00 6) lx3 f (x) , f ( x) = {22,+ ~~ ~~x2~~ ~~ , xx  33  \nx 2 − 1 x 2 − 5x  \n7) lim − ~~ ~~ 8) lim − ~~ ~~  \nx→ 1 x − 1 x→ 5 x − 5  \nx 2 − x − 2 x 2 + 3x − 10  \n9) lim − ~~ ~~ 10) lim ~~ ~~  \nx→ 2 x − 2 x→ −5 x + 5  \n 1  +  1  12) lim  x   \n−4 + x 4  x→ −3  1   1  \n11) lim −  \nx→ 0 x 3 + x 3  \n x − 5  x + 6 − 3  \n13) lim 14) lim ~~ ~~  \nx→ 5 x + 4 − 3 x→ 3 x − 3  \nCritical thinking questions:  \n15) Give an example of a limit of a rational function 16) Give two values of a where the limit cannot be where the limit at-1 exists, but the rational solved using direct evaluation. Give one value  \nfunction is undefined at-1. of a where the limit can be solved using direct  \nevaluation.  \n x   \nlim  \nx→ a 1 1  \n~~ ~~ + ~~ ~~  \n−2 + x 2  \nKuta Software-Infinite Calculus Name   \nEvaluating Limits Date  Period   \nEvaluate each limit.  \n1) lx f (x) , f (x) = {−x52, + 2, xx  22  \n−2  \nx 2 − 4 2) lim − ~~ ~~  \nx→ −2 x + 2  \n4  \nEvaluate each limit. You may use the provided graph to sketch the function.  \n3) lim x→ 3  \nx 2 − 7x + 12  \nx − 3  \n−1  \nEvaluate each limit.  \n5) lx f (x) , f (x) = {x2,+ 1, xx  00 1  \nx 2 − 1  \n7) lim − ~~  ~~x→ 1 x − 1  \n−2  \nx 2 − x − 2  \n9) lim − ~~  ~~x→ 2 x − 2  \n−3  \n 1  +  1  \n11) lim  −4 + x 4  x→ 0 x  \n 1   \n−  \n16  \n13) lim  x − 5   \nx→ 5 x + 4 − 3  \n6  \nCritical thinking questions:  \n15) Give an example of a limit of a rational function where the limit at-1 exists, but the rational function is un","cbCaicYM2BEk3i9Z","https://ap.wps.com/l/cbCaicYM2BEk3i9Z","pdf",44401,"English","# Evaluate each limit\n## Use direct evaluation or graphing as needed\n# Critical thinking questions","[{\"question\":\"When can a limit of a rational function be solved using direct evaluation?\",\"answer\":\"When the function is defined at the target x-value, direct substitution can be used to compute the limit. The worksheet includes prompts that ask for an example value of a where this applies.\"},{\"question\":\"What does it mean for a limit to exist even if the function is undefined at a point?\",\"answer\":\"The limit can still exist when the left-hand and right-hand values approach the same number, even though the function has no defined value at that point. The critical thinking question explicitly targets this situation at x = -1.\"},{\"question\":\"How should the graph be used in limits problems on this worksheet?\",\"answer\":\"For the problems that mention using the provided graph, sketch the behavior of the function near the indicated x-value(s). This helps determine the approaching y-value(s) when direct evaluation is not straightforward.\"}]","Evaluating Limits -  Limits at Removable Discontinuities - Worksheet | PDF",1791426192]