{"id":14865,"date":"2022-08-16T10:06:19","date_gmt":"2022-08-16T02:06:19","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=14865"},"modified":"2026-06-08T16:44:42","modified_gmt":"2026-06-08T08:44:42","slug":"chapter2-limits-and-derivatives-2-2-the-limit-of-a-function-2-4-the-precise-definition-of-a-limit","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-science\/department-of-applied-mathematics\/calculus-i-english-department-of-applied-mathematics-prof-michael-fuchs\/chapter2-limits-and-derivatives-2-2-the-limit-of-a-function-2-4-the-precise-definition-of-a-limit","title":{"rendered":"Chapter2 Limits and Derivatives, 2-2 The Limit of a Function, 2-4 The Precise Definition of a Limit"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"14865\" class=\"elementor elementor-14865\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky 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elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62afa6e2 elementor-widget elementor-widget-penci-info-box\" data-id=\"62afa6e2\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_95690\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t\t\t\t\t\t<h3 class=\"penci-ibox-title\">Chapter2 Limits and Derivatives, 2-2 The Limit of a Function, 2-4 The Precise Definition of a Limit\n<\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div 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to Course Homepage<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-723e657b\" data-id=\"723e657b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-336203b elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"336203b\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-73059f3c elementor-widget elementor-widget-shortcode\" data-id=\"73059f3c\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-24\" class=\"tablepress tablepress-id-24\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">Week<\/th><th class=\"column-2\">Course Content<\/th><th class=\"column-3\">Course Video<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter1 Functions and Model<br \/>\n1-5 Exponential Functions  <br \/>\n1-6 Inverse Functions and Logarithms<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14860&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter2 Limits and Derivatives<br \/>\n2-2 The Limit of a Function <br \/>\n2-4 The Precise Definition of a Limit<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14865&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2-3 Calculating Limits Using the Limit Laws<br \/>\n2-6 Limits at Infinity: Horizontal Asymptotes<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14874&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2-5 Continuity<br \/>\n2-6 Limits at Infinity; Horizontal Asymptotes<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14880&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2-8 Derivatives <br \/>\n2-9 The Derivative as a Function<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14886&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter3 Differentiation Rules<br \/>\n3-1 Derivatives of Polynomials and Exponential Functions <br \/>\n3-2 The Product and Quotient Rules <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14892&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3-4 Derivatives of Trigonometric Functions<br \/>\n3-5 The Chain Rule<br \/>\n3-6 Implicit Differentiation<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14898&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3-6 Implicit Differentiation<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14904&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3-8 Derivatives of Logarithmic Functions<br \/>\n3-10 Related Rates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14910&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3-7 Higher Derivatives <br \/>\n3-11 Linear Approximations and Differentials<br \/>\nChapter4 Applications of Differation<br \/>\n4-1 Maximum and Minimum Values<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14916&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-1 Maximum and Minimum Values<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14922&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-2 The Mean Value Theorem<br \/>\n4-3 How Derivatives Affect the Shape of a Graph<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14928&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-4 Indeterminate Forms a nd L\u2019Hospital\u2019s Rule<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14934&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-7 Optimization Problems<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14940&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-5 Summary of Curve Sketching<br \/>\n4-7 Optimization Problems<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14946&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">4-7 Optimization Problems<br \/>\n4-10 Antiderivatives<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14952&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter5 Integrals<br \/>\n5-1 Areas and Distances <br \/>\n5-2 The Definite Integral<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14961&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5-2 The Definite Integral<br \/>\n5-3 The Fundamental Theorem of Calculus<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14966&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5-4 Indefinite Integrals and the Total Change Theorem<br \/>\n5-5 The Substitution Rule<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14972&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">5-5 The Substitution Rule  <br \/>\n5-6 The Logarithm Defined as an Integral<br \/>\nChapter6 Applications of Integration<br \/>\n6-1 Areas between Curves<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14978&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6-1 Areas between Curves <br \/>\n6-2 Volumes<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14984&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">6-3 Volumes be Cylindrical Shells<br \/>\nChapter7 Techniques of Integration<br \/>\n7-1 Integration by Parts <br \/>\n7-2 Trigonometric Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14990&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7-2 Trigonometric Integrals<br \/>\n7-3 Trigonometric Substitution<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14996&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7-4 Integration of Rational Functions by Partial Fractions <br \/>\n7-8 Improper Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15002&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7-8 Improper Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15011&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">7-7 Approximate Integration <br \/>\nChapter8 Further Applications of Integration<br \/>\n8-1 Arc Length<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15018&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-28 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">8-1 Arc Length <br \/>\n8-2 Area of a Surface of Revolution<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15024&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-29 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter10 Parametric Equations and Polar Coordinates<br \/>\n10-1 Curves Defined by Parametric Equations <br \/>\n10-2 Calculus with Parametric Curves<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15030&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-30 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10-2 Calculus with Parametric Curves  <br \/>\n10-3 Polar Coordinates <br \/>\n10-4 Areas and Lengths in Polar Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15036&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-31 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">10-3 Polar Coordinates<br \/>\n10-4 Areas and Lengths in Polar Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=15042&amp;lang=en\">Watch Online<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-24 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-48a0b44\" data-id=\"48a0b44\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62a7c21 elementor-widget-divider--view-line_text elementor-widget-divider--element-align-center elementor-widget elementor-widget-divider\" data-id=\"62a7c21\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t\t<span 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tabindex=\"-1\">College of Computer Science<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5374\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-biological-science-and-technology\" class=\"elementor-item\" tabindex=\"-1\">College of Engineering Bioscience<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5379\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-humanities-and-social-sciences\" class=\"elementor-item\" tabindex=\"-1\">College of Humanities Arts and Social Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-5378\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-hakka-affair-culture\" class=\"elementor-item\" tabindex=\"-1\">College of Hakka Affair Culture<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118070\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/school-of-law\" class=\"elementor-item\" tabindex=\"-1\">School of Law<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118069\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/school-of-nursing-2\" class=\"elementor-item\" tabindex=\"-1\">School of Nursing<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118071\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-photonics\" class=\"elementor-item\" tabindex=\"-1\">College of Photonics<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-125874\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-biomedical-science-and-engineering\" class=\"elementor-item\" tabindex=\"-1\">College of Biomedical Science and Engineering<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-132738\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-life-sciences-en\" class=\"elementor-item\" tabindex=\"-1\">College of Life Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-141797\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/colege-of-medicine-en\" class=\"elementor-item\" tabindex=\"-1\">College of Medicine<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118072\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-chinese-center\" class=\"elementor-item\" tabindex=\"-1\">Course of Chinese Center<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120431\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/liberal-arts-college\" class=\"elementor-item\" tabindex=\"-1\">Forum Of Liberal Arts<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123746\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/speech-of-general-education-center\" class=\"elementor-item\" tabindex=\"-1\">Contemporary Civilization<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128240\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/reading-of-classic-masterpiece\" class=\"elementor-item\" tabindex=\"-1\">Reading of Classic Masterpiece<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123745\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-general-education-center-2\" class=\"elementor-item\" tabindex=\"-1\">Course of General Education Center<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Chapter2 Limit&hellip;<\/p>\n","protected":false},"featured_media":0,"parent":14657,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","format":"standard","meta":{"footnotes":""},"categories":[898],"tags":[],"class_list":["post-14865","course_page","type-course_page","status-publish","format-standard","hentry","category-calculus-i-english-96-en"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Chapter2 Limits and Derivatives, 2-2 The Limit of a Function, 2-4 The Precise Definition of a Limit - 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