{"id":3405,"date":"2024-06-24T16:29:48","date_gmt":"2024-06-24T16:29:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3405"},"modified":"2024-08-05T12:22:57","modified_gmt":"2024-08-05T12:22:57","slug":"introduction-to-the-limit-of-a-function-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-the-limit-of-a-function-apply-it\/","title":{"raw":"Introduction to the Limit of a Function: Apply It","rendered":"Introduction to the Limit of a Function: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs<\/li>\r\n\t<li>Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits<\/li>\r\n\t<li>Understand and use the proper notation for infinite limits and define vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The number [latex]e[\/latex] is a constant that, like [latex]\\pi[\/latex], is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not the root of a non-zero polynomial of finite degree with rational coefficients). It was first discovered by the Swiss mathematician <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jacob_Bernoulli\">Jacob Bernoulli<\/a> in 1683 while studying compound interest. It has many real-world applications including <a href=\"https:\/\/en.wikipedia.org\/wiki\/Probability_theory\">probability<\/a> <a href=\"https:\/\/en.wikipedia.org\/wiki\/Probability_theory\">theory<\/a> and exponential growth and decay. Below is an approximation of [latex]e[\/latex] to [latex]20[\/latex] decimal places.<\/p>\r\n<p style=\"text-align: center;\">[latex]e \\approx 2.71828182845904523536[\/latex]<\/p>\r\n<p>This number is perhaps best defined by a limit. Consider the function [latex]f(x)=(1+x)^{\\frac{1}{x}}[\/latex]. We cannot compute [latex]f(0)[\/latex], but we can see what happens with values of [latex]x[\/latex] close to zero.<\/p>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287754[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287755[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287756[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287757[\/ohm_question]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs<\/li>\n<li>Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits<\/li>\n<li>Understand and use the proper notation for infinite limits and define vertical asymptotes<\/li>\n<\/ul>\n<\/section>\n<p>The number [latex]e[\/latex] is a constant that, like [latex]\\pi[\/latex], is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not the root of a non-zero polynomial of finite degree with rational coefficients). It was first discovered by the Swiss mathematician <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jacob_Bernoulli\">Jacob Bernoulli<\/a> in 1683 while studying compound interest. It has many real-world applications including <a href=\"https:\/\/en.wikipedia.org\/wiki\/Probability_theory\">probability<\/a> <a href=\"https:\/\/en.wikipedia.org\/wiki\/Probability_theory\">theory<\/a> and exponential growth and decay. Below is an approximation of [latex]e[\/latex] to [latex]20[\/latex] decimal places.<\/p>\n<p style=\"text-align: center;\">[latex]e \\approx 2.71828182845904523536[\/latex]<\/p>\n<p>This number is perhaps best defined by a limit. Consider the function [latex]f(x)=(1+x)^{\\frac{1}{x}}[\/latex]. We cannot compute [latex]f(0)[\/latex], but we can see what happens with values of [latex]x[\/latex] close to zero.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287754\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287754&theme=lumen&iframe_resize_id=ohm287754&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287755\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287755&theme=lumen&iframe_resize_id=ohm287755&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287756\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287756&theme=lumen&iframe_resize_id=ohm287756&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287757\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287757&theme=lumen&iframe_resize_id=ohm287757&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":484,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3405"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3405\/revisions"}],"predecessor-version":[{"id":4465,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3405\/revisions\/4465"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/484"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3405\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3405"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3405"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3405"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}