{"id":3304,"date":"2025-08-16T01:54:17","date_gmt":"2025-08-16T01:54:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3304"},"modified":"2026-02-24T21:28:35","modified_gmt":"2026-02-24T21:28:35","slug":"finding-limits-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-apply-it\/","title":{"raw":"Finding Limits: Apply It","rendered":"Finding Limits: Apply It"},"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<p class=\"whitespace-normal break-words\">A car's speedometer shows instantaneous speed\u2014how fast you're traveling at a specific moment. But how do we calculate instantaneous velocity from position data? Limits provide the answer by examining what happens as time intervals become infinitesimally small.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">Average velocity over an interval is calculated as [latex]\\frac{\\text{change in position}}{\\text{change in time}}[\/latex].<\/section>\r\n<p class=\"whitespace-normal break-words\">Instantaneous velocity is found by taking the limit as the time interval approaches zero\u2014this is the foundation of derivatives in calculus.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A factory monitors the temperature [latex]T(t)[\/latex] in degrees Celsius of a chemical reaction over time [latex]t[\/latex] in minutes. The graph shows the temperature function with a break at [latex]t = 3[\/latex] minutes when operators adjust the reaction conditions.<\/p>\r\n<p class=\"whitespace-normal break-words\"><img class=\"alignnone wp-image-4883\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature.png\" alt=\"Line with consistent positive slope breaks at (3,75) with an open circle. Starts up again at (3,60) with no circle.\" width=\"448\" height=\"361\" \/><\/p>\r\n<p class=\"whitespace-normal break-words\">Use the graph to find:<\/p>\r\n<p class=\"whitespace-normal break-words\">a) [latex]\\lim_{t \\to 3^-} T(t)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">b) [latex]\\lim_{t \\to 3^+} T(t)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">c) [latex]\\lim_{t \\to 3} T(t)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">d) [latex]T(3)[\/latex]<\/p>\r\n[reveal-answer q=\"249439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"249439\"]\r\na) Left-hand limit: As [latex]t[\/latex] approaches 3 from the left ([latex]t &lt; 3[\/latex]), we observe the branch of the graph to the left of [latex]t = 3[\/latex]. The temperature values approach 75\u00b0C. [latex]\\lim_{t \\to 3^-} T(t) = 75[\/latex]\r\n\r\nb) Right-hand limit: As [latex]t[\/latex] approaches 3 from the right ([latex]t &gt; 3[\/latex]), we observe the branch to the right of [latex]t = 3[\/latex]. The temperature values approach 60\u00b0C. [latex]\\lim_{t \\to 3^+} T(t) = 60[\/latex]\r\n\r\nc) Two-sided limit: Since the left-hand limit (75) does not equal the right-hand limit (60), the two-sided limit does not exist. [latex]\\lim_{t \\to 3} T(t) \\text{ does not exist}[\/latex]\r\n\r\nd) Function value: The filled circle at [latex](3, 60)[\/latex] indicates the actual temperature at [latex]t = 3[\/latex]. [latex]T(3) = 60[\/latex]\u00b0C[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">A limit can exist even when the function value doesn't, and a function value can exist even when the limit doesn't! The limit only cares about what's happening <em>near<\/em> the point, not <em>at<\/em> the point.<\/section><section class=\"textbox example\" aria-label=\"Example\">A population biologist models bacterial growth with the function [latex]f(x) = \\frac{x^3 - 125}{x - 5}[\/latex], where [latex]x[\/latex] represents hours after midnight. Estimate [latex]\\lim_{x \\to 5} f(x)[\/latex] using a table of values.[reveal-answer q=\"725273\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725273\"]Notice that [latex]f(5)[\/latex] is undefined because substituting [latex]x = 5[\/latex] gives [latex]\\frac{0}{0}[\/latex]. However, we can still find the limit by examining values near 5. Let's create a table with values approaching 5 from both sides: [latex]x[\/latex] (from left) [latex]f(x)[\/latex] [latex]x[\/latex] (from right) [latex]f(x)[\/latex] 4.9 73.51 5.1 76.51 4.99 74.8501 5.01 75.1501 4.999 74.985 5.001 75.015 4.9999 74.9985 5.0001 75.0015 Analysis: As [latex]x[\/latex] approaches 5 from the left, [latex]f(x)[\/latex] approaches 75. As [latex]x[\/latex] approaches 5 from the right, [latex]f(x)[\/latex] approaches 75. Since both one-sided limits equal 75: [latex]\\lim_{x \\to 5} f(x) = 75[\/latex] Even though [latex]f(5)[\/latex] doesn't exist, the bacterial population model predicts the population is approaching a value corresponding to 75 at the 5-hour mark.[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"42\" data-end=\"252\">A ball is dropped from a height, and the function<br data-start=\"91\" data-end=\"94\" \/>[latex]h(t) = \\dfrac{5t^{2}-3t+2}{t-1}[\/latex]<br data-start=\"140\" data-end=\"143\" \/>models the height of the ball (in meters) at time [latex]t[\/latex] seconds, except when [latex]t=1[\/latex].<\/p>\r\n<p data-start=\"254\" data-end=\"338\">Use a table of values to estimate [latex]\\displaystyle \\lim_{t\\to 1} h(t)[\/latex].<\/p>\r\n\r\n<div class=\"_tableContainer_1rjym_1\">\r\n<div class=\"group _tableWrapper_1rjym_13 flex w-fit flex-col-reverse\" tabindex=\"-1\">\r\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"340\" data-end=\"480\">\r\n<thead data-start=\"340\" data-end=\"385\">\r\n<tr data-start=\"340\" data-end=\"385\">\r\n<th data-start=\"340\" data-end=\"344\" data-col-size=\"sm\">t<\/th>\r\n<th data-start=\"344\" data-end=\"350\" data-col-size=\"sm\">0.9<\/th>\r\n<th data-start=\"350\" data-end=\"357\" data-col-size=\"sm\">0.95<\/th>\r\n<th data-start=\"357\" data-end=\"364\" data-col-size=\"sm\">0.99<\/th>\r\n<th data-start=\"364\" data-end=\"371\" data-col-size=\"sm\">1.01<\/th>\r\n<th data-start=\"371\" data-end=\"378\" data-col-size=\"sm\">1.05<\/th>\r\n<th data-start=\"378\" data-end=\"385\" data-col-size=\"sm\">1.1<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody data-start=\"432\" data-end=\"480\">\r\n<tr data-start=\"432\" data-end=\"480\">\r\n<td data-start=\"432\" data-end=\"439\" data-col-size=\"sm\">h(t)<\/td>\r\n<td data-col-size=\"sm\" data-start=\"439\" data-end=\"445\"><\/td>\r\n<td data-col-size=\"sm\" data-start=\"445\" data-end=\"452\"><\/td>\r\n<td data-col-size=\"sm\" data-start=\"452\" data-end=\"459\"><\/td>\r\n<td data-col-size=\"sm\" data-start=\"459\" data-end=\"466\"><\/td>\r\n<td data-col-size=\"sm\" data-start=\"466\" data-end=\"473\"><\/td>\r\n<td data-col-size=\"sm\" data-start=\"473\" data-end=\"480\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<p data-start=\"482\" data-end=\"574\" data-is-last-node=\"\" data-is-only-node=\"\">Estimate the value that [latex]h(t)[\/latex] approaches as [latex]t[\/latex] gets closer to 1.<\/p>\r\n\r\n<\/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<p class=\"whitespace-normal break-words\">A car&#8217;s speedometer shows instantaneous speed\u2014how fast you&#8217;re traveling at a specific moment. But how do we calculate instantaneous velocity from position data? Limits provide the answer by examining what happens as time intervals become infinitesimally small.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">Average velocity over an interval is calculated as [latex]\\frac{\\text{change in position}}{\\text{change in time}}[\/latex].<\/section>\n<p class=\"whitespace-normal break-words\">Instantaneous velocity is found by taking the limit as the time interval approaches zero\u2014this is the foundation of derivatives in calculus.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A factory monitors the temperature [latex]T(t)[\/latex] in degrees Celsius of a chemical reaction over time [latex]t[\/latex] in minutes. The graph shows the temperature function with a break at [latex]t = 3[\/latex] minutes when operators adjust the reaction conditions.<\/p>\n<p class=\"whitespace-normal break-words\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4883\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature.png\" alt=\"Line with consistent positive slope breaks at (3,75) with an open circle. Starts up again at (3,60) with no circle.\" width=\"448\" height=\"361\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-300x242.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-1024x827.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-768x620.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-65x52.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-225x182.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23193702\/Limit-Temperature-350x283.png 350w\" sizes=\"(max-width: 448px) 100vw, 448px\" \/><\/p>\n<p class=\"whitespace-normal break-words\">Use the graph to find:<\/p>\n<p class=\"whitespace-normal break-words\">a) [latex]\\lim_{t \\to 3^-} T(t)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">b) [latex]\\lim_{t \\to 3^+} T(t)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">c) [latex]\\lim_{t \\to 3} T(t)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">d) [latex]T(3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q249439\">Show Solution<\/button><\/p>\n<div id=\"q249439\" class=\"hidden-answer\" style=\"display: none\">\na) Left-hand limit: As [latex]t[\/latex] approaches 3 from the left ([latex]t < 3[\/latex]), we observe the branch of the graph to the left of [latex]t = 3[\/latex]. The temperature values approach 75\u00b0C. [latex]\\lim_{t \\to 3^-} T(t) = 75[\/latex]\n\nb) Right-hand limit: As [latex]t[\/latex] approaches 3 from the right ([latex]t > 3[\/latex]), we observe the branch to the right of [latex]t = 3[\/latex]. The temperature values approach 60\u00b0C. [latex]\\lim_{t \\to 3^+} T(t) = 60[\/latex]<\/p>\n<p>c) Two-sided limit: Since the left-hand limit (75) does not equal the right-hand limit (60), the two-sided limit does not exist. [latex]\\lim_{t \\to 3} T(t) \\text{ does not exist}[\/latex]<\/p>\n<p>d) Function value: The filled circle at [latex](3, 60)[\/latex] indicates the actual temperature at [latex]t = 3[\/latex]. [latex]T(3) = 60[\/latex]\u00b0C<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">A limit can exist even when the function value doesn&#8217;t, and a function value can exist even when the limit doesn&#8217;t! The limit only cares about what&#8217;s happening <em>near<\/em> the point, not <em>at<\/em> the point.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A population biologist models bacterial growth with the function [latex]f(x) = \\frac{x^3 - 125}{x - 5}[\/latex], where [latex]x[\/latex] represents hours after midnight. Estimate [latex]\\lim_{x \\to 5} f(x)[\/latex] using a table of values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q725273\">Show Solution<\/button><\/p>\n<div id=\"q725273\" class=\"hidden-answer\" style=\"display: none\">Notice that [latex]f(5)[\/latex] is undefined because substituting [latex]x = 5[\/latex] gives [latex]\\frac{0}{0}[\/latex]. However, we can still find the limit by examining values near 5. Let&#8217;s create a table with values approaching 5 from both sides: [latex]x[\/latex] (from left) [latex]f(x)[\/latex] [latex]x[\/latex] (from right) [latex]f(x)[\/latex] 4.9 73.51 5.1 76.51 4.99 74.8501 5.01 75.1501 4.999 74.985 5.001 75.015 4.9999 74.9985 5.0001 75.0015 Analysis: As [latex]x[\/latex] approaches 5 from the left, [latex]f(x)[\/latex] approaches 75. As [latex]x[\/latex] approaches 5 from the right, [latex]f(x)[\/latex] approaches 75. Since both one-sided limits equal 75: [latex]\\lim_{x \\to 5} f(x) = 75[\/latex] Even though [latex]f(5)[\/latex] doesn&#8217;t exist, the bacterial population model predicts the population is approaching a value corresponding to 75 at the 5-hour mark.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"42\" data-end=\"252\">A ball is dropped from a height, and the function<br data-start=\"91\" data-end=\"94\" \/>[latex]h(t) = \\dfrac{5t^{2}-3t+2}{t-1}[\/latex]<br data-start=\"140\" data-end=\"143\" \/>models the height of the ball (in meters) at time [latex]t[\/latex] seconds, except when [latex]t=1[\/latex].<\/p>\n<p data-start=\"254\" data-end=\"338\">Use a table of values to estimate [latex]\\displaystyle \\lim_{t\\to 1} h(t)[\/latex].<\/p>\n<div class=\"_tableContainer_1rjym_1\">\n<div class=\"group _tableWrapper_1rjym_13 flex w-fit flex-col-reverse\" tabindex=\"-1\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"340\" data-end=\"480\">\n<thead data-start=\"340\" data-end=\"385\">\n<tr data-start=\"340\" data-end=\"385\">\n<th data-start=\"340\" data-end=\"344\" data-col-size=\"sm\">t<\/th>\n<th data-start=\"344\" data-end=\"350\" data-col-size=\"sm\">0.9<\/th>\n<th data-start=\"350\" data-end=\"357\" data-col-size=\"sm\">0.95<\/th>\n<th data-start=\"357\" data-end=\"364\" data-col-size=\"sm\">0.99<\/th>\n<th data-start=\"364\" data-end=\"371\" data-col-size=\"sm\">1.01<\/th>\n<th data-start=\"371\" data-end=\"378\" data-col-size=\"sm\">1.05<\/th>\n<th data-start=\"378\" data-end=\"385\" data-col-size=\"sm\">1.1<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"432\" data-end=\"480\">\n<tr data-start=\"432\" data-end=\"480\">\n<td data-start=\"432\" data-end=\"439\" data-col-size=\"sm\">h(t)<\/td>\n<td data-col-size=\"sm\" data-start=\"439\" data-end=\"445\"><\/td>\n<td data-col-size=\"sm\" data-start=\"445\" data-end=\"452\"><\/td>\n<td data-col-size=\"sm\" data-start=\"452\" data-end=\"459\"><\/td>\n<td data-col-size=\"sm\" data-start=\"459\" data-end=\"466\"><\/td>\n<td data-col-size=\"sm\" data-start=\"466\" data-end=\"473\"><\/td>\n<td data-col-size=\"sm\" data-start=\"473\" data-end=\"480\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p data-start=\"482\" data-end=\"574\" data-is-last-node=\"\" data-is-only-node=\"\">Estimate the value that [latex]h(t)[\/latex] approaches as [latex]t[\/latex] gets closer to 1.<\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3304"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3304\/revisions"}],"predecessor-version":[{"id":5730,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3304\/revisions\/5730"}],"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\/3304\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3304"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3304"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3304"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3304"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}