[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-523583-105":59,"doc-detail-523583-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","limits-by-direct-evaluation-precalculus-worksheet","Limits by Direct Evaluation - Precalculus Worksheet","","This document provides a set of precalculus practice problems focused on evaluating limits by direct substitution. It features ten distinct algebraic and trigonometric limit problems, along with two critical thinking questions that challenge students to construct their own limit expressions for specific results. Designed as a study aid for precalculus students, it helps in mastering the fundamental techniques of limit evaluation in mathematical analysis, offering clear examples for practice and self-assessment.",{"@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/limits-by-direct-evaluation-precalculus-worksheet/523583/",{"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/limits-by-direct-evaluation-precalculus-worksheet/523583.png","ImageObject",300,407,{"name":92,"@type":93},"MrHarris58","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the primary objective of this worksheet?","Question",{"text":112,"@type":113},"The worksheet aims to help students practice evaluating limits using the method of direct evaluation.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Are there any trigonometric limits included in the exercises?",{"text":117,"@type":113},"Yes, question 10 requires students to evaluate the limit of a cosine function.",{"name":119,"@type":110,"acceptedAnswer":120},"Do the critical thinking questions require specific answers?",{"text":121,"@type":113},"No, the document notes that there are many possible answers for the critical thinking questions, providing example solutions for each.","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},523583,1791534481,{"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":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"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":24,"preview_page_count":4},687212988360,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Kuta Software-Infinite Precalculus Name   \nEvaluating Limits Date  Period   \nEvaluate each limit.  \n1) im􀁯 􀀓 􀀐􀀔 2) im􀁯 􀀐􀀗 (􀀐x 􀀐 􀀗)  \n3) im􀁯 􀀐􀀕 (􀀕x 􀀎 􀀖) 4) 􀀖 (􀀐x􀀖 􀀎 􀀔􀀖 x 􀀕 􀀐 􀀘􀀙x 􀀎 􀀛􀀔)  \n5) im􀁯 􀀕 (x 􀀖 􀀐 􀀕x 􀀕 􀀐 􀀕) 6) 􀀘 􀀐 􀀕x 􀀎 􀀔  \n7) im􀁯 􀀐􀀕 􀀖 􀀐􀀕x 􀀐 􀀖 8) im􀁯 􀀐􀀖 x~~ ~~􀀕~~ ~~x􀀎~~ ~~x􀀖􀀎~~ ~~􀀕~~ ~~  \nx 􀀐 􀀙 10) lim cos (x)  \n9) im􀁯 􀀕 x 􀀕 􀀐 􀀗x 􀀎 􀀖  x􀁯 ~~ ~~  \nCritical thinking questions:  \n11) Give an example of a limit that evaluates to 12) Give an example of a limit of a quadratic  \n5. function where the limit evaluates to 16.  \nKuta Software-Infinite Precalculus Evaluating Limits  \nEvaluate each limit.  \n1)  \n3)  \n5)  \n7)  \n9)  \nim􀁯 􀀓 􀀐􀀔  \n􀀐􀀔  \nlim  \nx􀁯 􀀐􀀕  \n(􀀕x 􀀎 􀀖)  \n􀀐􀀔  \nim􀁯 􀀕 (x 􀀖 􀀐 􀀕x 􀀕 􀀐 􀀕)  \n􀀐􀀕  \nlim  \nx􀁯 􀀐􀀕  \n 􀀖 􀀐􀀕x 􀀐 􀀖   \n􀀔  \nim􀁯 􀀕 x~~ 􀀕 ~~~~ ~~x􀀙􀀎~~ ~~􀀖~~ ~~􀀗  \nName  Date  Period   \n2)  \n4)  \n6)  \n8)  \nlim  \nx􀁯 􀀐􀀗  \n(􀀐x 􀀐 􀀗)  \n􀀓  \n􀀖   \n􀀖  \n(􀀐x􀀖 􀀎 􀀔􀀖 x 􀀕 􀀐 􀀘􀀙x 􀀎 􀀛􀀔)  \nlim 􀀐 􀀕x 􀀎 􀀔 x􀁯 􀀘  \n􀀐 􀀔􀀔  \nx 􀀐 􀀖 im􀁯 􀀐􀀖 x 􀀕 􀀎 􀀕x 􀀎 􀀕   \n􀀙 􀀐  \n􀀘  \n10) lim cos (x)  \n􀁓 x􀁯  \n􀀙  \n􀀖  \n􀀕  \nCritical thinking questions:  \n11) Give an example of a limit that evaluates to 5.  \nMany answers. Ex:  \nlim x  \nx􀁯 􀀘  \n12) Give an example of a limit of a quadratic function where the limit evaluates to 16.  \nMany answers. Ex:  \nlim  \nx􀁯 􀀗  \n􀀕  \nx  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaia9HamUKOHcJ","https://ap.wps.com/l/cbCaia9HamUKOHcJ","pdf",22732,"English","# Practice Problems\n## Critical Thinking Questions","[{\"question\":\"What is the primary objective of this worksheet?\",\"answer\":\"The worksheet aims to help students practice evaluating limits using the method of direct evaluation.\"},{\"question\":\"Are there any trigonometric limits included in the exercises?\",\"answer\":\"Yes, question 10 requires students to evaluate the limit of a cosine function.\"},{\"question\":\"Do the critical thinking questions require specific answers?\",\"answer\":\"No, the document notes that there are many possible answers for the critical thinking questions, providing example solutions for each.\"}]","Limits by Direct Evaluation - Precalculus Worksheet | PDF",1791427065]