{"id":1607,"date":"2025-07-25T04:05:45","date_gmt":"2025-07-25T04:05:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1607"},"modified":"2026-04-03T17:27:14","modified_gmt":"2026-04-03T17:27:14","slug":"finding-limits-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-fresh-take\/","title":{"raw":"Finding Limits: Fresh Take","rendered":"Finding Limits: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find a limit using a graph<\/li>\r\n \t<li>Find a limit using a table<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Find a Limit Using a Graph<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>A limit describes what value a function approaches as the input (x) gets closer to some point. Graphs give a visual way to estimate limits. You check how the function behaves from both the left side and the right side of the point of interest. If both sides approach the same y-value, that's the limit.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Graphs to Find Limits<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li><strong>Locate the Point of Interest<\/strong>\r\n<ul>\r\n \t<li>Identify the x-value you're approaching, say [latex]x=a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Check from the Left ([latex]x \\to a^{-}[\/latex])<\/strong>\r\n<ul>\r\n \t<li>Trace along the curve from values smaller than [latex]a[\/latex]<\/li>\r\n \t<li>See what y-value the graph is approaching<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Check from the Right ([latex]x \\to a^{+}[\/latex])<\/strong>\r\n<ul>\r\n \t<li>Trace from values larger than [latex]a[\/latex]<\/li>\r\n \t<li>See what y-value the graph is approaching<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Compare the Two Sides<\/strong>\r\n<ul>\r\n \t<li>If left-hand and right-hand values match, the limit exists and equals that y-value<\/li>\r\n \t<li>If they don't match, the limit does not exist<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Special Case: Holes in the Graph<\/strong>\r\n<ul>\r\n \t<li>Even if the function is not defined at [latex]x=a[\/latex], the limit can still exist<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Use the graph below to find [latex]\\lim_{x\\to 2} f(x)[\/latex].\r\n<img class=\"alignnone wp-image-6019\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM.png\" alt=\"A rational function is graphed with two branches and a vertical asymptote at x = 1. For x greater than 1, the curve is high near the asymptote, then decreases as x increases. There is an open circle at approximately (2, 3) indicating a hole in the graph. The curve continues decreasing and levels off toward a horizontal value slightly above y = 1. For x less than 1, the curve lies below the x-axis for values near the asymptote and drops toward negative infinity as it approaches x = 1 from the left. As x decreases further, the graph rises, crossing the x-axis near x \u2248 \u22121, and continues increasing slowly.\" width=\"194\" height=\"188\" \/>[reveal-answer q=\"graph-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"graph-001\"]From the graph:Approaching from the left ([latex]x \\to 2^{-}[\/latex]):\r\nAs x-values get closer to 2 from the left (1.9, 1.99, 1.999...), the y-values approach 3.\r\n\r\nApproaching from the right ([latex]x \\to 2^{+}[\/latex]):\r\nAs x-values get closer to 2 from the right (2.1, 2.01, 2.001...), the y-values also approach 3.Since both the left-hand and right-hand limits equal 3,\r\n\r\n[latex]\\lim_{x\\to 2} f(x) = 3[\/latex]\r\n\r\nNote: Even though there is a hole at [latex]x = 2[\/latex] (the function is not defined there), the limit still exists.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-caadgaeb-Vi4BiJj-n0g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vi4BiJj-n0g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-caadgaeb-Vi4BiJj-n0g\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14835596&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-caadgaeb-Vi4BiJj-n0g&vembed=0&video_id=Vi4BiJj-n0g&video_target=tpm-plugin-caadgaeb-Vi4BiJj-n0g'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+2+-+Determine+Limits+from+a+Given+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Determine Limits from a Given Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Find a Limit Using a Table<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nA limit describes what value a function approaches as [latex]x[\/latex] gets closer to some point, even if the function isn't defined there. A table of values helps approximate limits numerically: plug in values of [latex]x[\/latex] near the point from both sides and look at the corresponding [latex]f(x)[\/latex] values. If the y-values get closer to the same number from both sides, that number is the limit.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Tables to Find Limits<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li><strong>Pick the Point of Interest<\/strong>\r\n<ul>\r\n \t<li>Suppose we want [latex]\\lim_{x\\to a} f(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Choose x-Values Close to a<\/strong>\r\n<ul>\r\n \t<li>Select values just smaller than [latex]a[\/latex] (left-hand approach)<\/li>\r\n \t<li>Select values just larger than [latex]a[\/latex] (right-hand approach)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Build the Table<\/strong>\r\n<ul>\r\n \t<li>Compute [latex]f(x)[\/latex] at each chosen value<\/li>\r\n \t<li>Organize so left-hand and right-hand values are easy to compare<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Look for a Pattern<\/strong>\r\n<ul>\r\n \t<li>If both sides approach the same number, that's the limit<\/li>\r\n \t<li>If left and right sides don't match, the limit does not exist<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Use a table to find [latex]\\lim_{x\\to 3} \\frac{x^2 - 9}{x - 3}[\/latex].[reveal-answer q=\"table-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"table-001\"]Build a table with values approaching 3 from both sides:[latex]\\begin{array}{|c|c|}\r\n\\hline\r\nx &amp; f(x) = \\frac{x^2-9}{x-3} \\\\\r\n\\hline\r\n2.9 &amp; 5.9 \\\\\r\n2.99 &amp; 5.99 \\\\\r\n2.999 &amp; 5.999 \\\\\r\n\\hline\r\n3.001 &amp; 6.001 \\\\\r\n3.01 &amp; 6.01 \\\\\r\n3.1 &amp; 6.1 \\\\\r\n\\hline\r\n\\end{array}[\/latex]From the left ([latex]x \\to 3^{-}[\/latex]):\r\nThe values 5.9, 5.99, 5.999 are approaching 6.From the right ([latex]x \\to 3^{+}[\/latex]):\r\nThe values 6.001, 6.01, 6.1 are also approaching 6.Since both sides approach 6,\r\n\r\n[latex]\\lim_{x\\to 3} \\frac{x^2-9}{x-3} = 6[\/latex]\r\n\r\nNote: Even though [latex]f(3)[\/latex] is undefined (division by zero), the limit exists and equals 6.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\">https:\/\/www.youtube.com\/watch?v=oZuGaurMbIk<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find a limit using a graph<\/li>\n<li>Find a limit using a table<\/li>\n<\/ul>\n<\/section>\n<h2>Find a Limit Using a Graph<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong>A limit describes what value a function approaches as the input (x) gets closer to some point. Graphs give a visual way to estimate limits. You check how the function behaves from both the left side and the right side of the point of interest. If both sides approach the same y-value, that&#8217;s the limit.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Graphs to Find Limits<\/strong><\/p>\n<ol>\n<li><strong>Locate the Point of Interest<\/strong>\n<ul>\n<li>Identify the x-value you&#8217;re approaching, say [latex]x=a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Check from the Left ([latex]x \\to a^{-}[\/latex])<\/strong>\n<ul>\n<li>Trace along the curve from values smaller than [latex]a[\/latex]<\/li>\n<li>See what y-value the graph is approaching<\/li>\n<\/ul>\n<\/li>\n<li><strong>Check from the Right ([latex]x \\to a^{+}[\/latex])<\/strong>\n<ul>\n<li>Trace from values larger than [latex]a[\/latex]<\/li>\n<li>See what y-value the graph is approaching<\/li>\n<\/ul>\n<\/li>\n<li><strong>Compare the Two Sides<\/strong>\n<ul>\n<li>If left-hand and right-hand values match, the limit exists and equals that y-value<\/li>\n<li>If they don&#8217;t match, the limit does not exist<\/li>\n<\/ul>\n<\/li>\n<li><strong>Special Case: Holes in the Graph<\/strong>\n<ul>\n<li>Even if the function is not defined at [latex]x=a[\/latex], the limit can still exist<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use the graph below to find [latex]\\lim_{x\\to 2} f(x)[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6019\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM.png\" alt=\"A rational function is graphed with two branches and a vertical asymptote at x = 1. For x greater than 1, the curve is high near the asymptote, then decreases as x increases. There is an open circle at approximately (2, 3) indicating a hole in the graph. The curve continues decreasing and levels off toward a horizontal value slightly above y = 1. For x less than 1, the curve lies below the x-axis for values near the asymptote and drops toward negative infinity as it approaches x = 1 from the left. As x decreases further, the graph rises, crossing the x-axis near x \u2248 \u22121, and continues increasing slowly.\" width=\"194\" height=\"188\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM.png 502w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM-300x290.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM-65x63.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM-225x218.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/25173813\/Screenshot-2026-03-25-at-10.37.39%E2%80%AFAM-350x339.png 350w\" sizes=\"(max-width: 194px) 100vw, 194px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qgraph-001\">Show Solution<\/button><\/p>\n<div id=\"qgraph-001\" class=\"hidden-answer\" style=\"display: none\">From the graph:Approaching from the left ([latex]x \\to 2^{-}[\/latex]):<br \/>\nAs x-values get closer to 2 from the left (1.9, 1.99, 1.999&#8230;), the y-values approach 3.<\/p>\n<p>Approaching from the right ([latex]x \\to 2^{+}[\/latex]):<br \/>\nAs x-values get closer to 2 from the right (2.1, 2.01, 2.001&#8230;), the y-values also approach 3.Since both the left-hand and right-hand limits equal 3,<\/p>\n<p>[latex]\\lim_{x\\to 2} f(x) = 3[\/latex]<\/p>\n<p>Note: Even though there is a hole at [latex]x = 2[\/latex] (the function is not defined there), the limit still exists.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-caadgaeb-Vi4BiJj-n0g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vi4BiJj-n0g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-caadgaeb-Vi4BiJj-n0g\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14835596&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-caadgaeb-Vi4BiJj-n0g&#38;vembed=0&#38;video_id=Vi4BiJj-n0g&#38;video_target=tpm-plugin-caadgaeb-Vi4BiJj-n0g\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+2+-+Determine+Limits+from+a+Given+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Determine Limits from a Given Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Find a Limit Using a Table<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>A limit describes what value a function approaches as [latex]x[\/latex] gets closer to some point, even if the function isn&#8217;t defined there. A table of values helps approximate limits numerically: plug in values of [latex]x[\/latex] near the point from both sides and look at the corresponding [latex]f(x)[\/latex] values. If the y-values get closer to the same number from both sides, that number is the limit.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Tables to Find Limits<\/strong><\/p>\n<ol>\n<li><strong>Pick the Point of Interest<\/strong>\n<ul>\n<li>Suppose we want [latex]\\lim_{x\\to a} f(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Choose x-Values Close to a<\/strong>\n<ul>\n<li>Select values just smaller than [latex]a[\/latex] (left-hand approach)<\/li>\n<li>Select values just larger than [latex]a[\/latex] (right-hand approach)<\/li>\n<\/ul>\n<\/li>\n<li><strong>Build the Table<\/strong>\n<ul>\n<li>Compute [latex]f(x)[\/latex] at each chosen value<\/li>\n<li>Organize so left-hand and right-hand values are easy to compare<\/li>\n<\/ul>\n<\/li>\n<li><strong>Look for a Pattern<\/strong>\n<ul>\n<li>If both sides approach the same number, that&#8217;s the limit<\/li>\n<li>If left and right sides don&#8217;t match, the limit does not exist<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Use a table to find [latex]\\lim_{x\\to 3} \\frac{x^2 - 9}{x - 3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qtable-001\">Show Solution<\/button><\/p>\n<div id=\"qtable-001\" class=\"hidden-answer\" style=\"display: none\">Build a table with values approaching 3 from both sides:[latex]\\begin{array}{|c|c|}  \\hline  x & f(x) = \\frac{x^2-9}{x-3} \\\\  \\hline  2.9 & 5.9 \\\\  2.99 & 5.99 \\\\  2.999 & 5.999 \\\\  \\hline  3.001 & 6.001 \\\\  3.01 & 6.01 \\\\  3.1 & 6.1 \\\\  \\hline  \\end{array}[\/latex]From the left ([latex]x \\to 3^{-}[\/latex]):<br \/>\nThe values 5.9, 5.99, 5.999 are approaching 6.From the right ([latex]x \\to 3^{+}[\/latex]):<br \/>\nThe values 6.001, 6.01, 6.1 are also approaching 6.Since both sides approach 6,<\/p>\n<p>[latex]\\lim_{x\\to 3} \\frac{x^2-9}{x-3} = 6[\/latex]<\/p>\n<p>Note: Even though [latex]f(3)[\/latex] is undefined (division by zero), the limit exists and equals 6.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">https:\/\/www.youtube.com\/watch?v=oZuGaurMbIk<\/section>\n","protected":false},"author":67,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Determine Limits from a Given Graph\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Vi4BiJj-n0g\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding a limit using a table\",\"author\":\"\",\"organization\":\"ProfessorNidiaGonzalez\",\"url\":\"https:\/\/youtu.be\/oZuGaurMbIk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex 2: Determine Limits from a Given Graph","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/Vi4BiJj-n0g","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Finding a limit using a table","author":"","organization":"ProfessorNidiaGonzalez","url":"https:\/\/youtu.be\/oZuGaurMbIk","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14835596&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-caadgaeb-Vi4BiJj-n0g&vembed=0&video_id=Vi4BiJj-n0g&video_target=tpm-plugin-caadgaeb-Vi4BiJj-n0g'><\/script>\n","media_targets":["tpm-plugin-caadgaeb-Vi4BiJj-n0g"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1607"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1607\/revisions"}],"predecessor-version":[{"id":6108,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1607\/revisions\/6108"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1607\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1607"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1607"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1607"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1607"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}