{"id":221488,"date":"2025-05-29T17:21:15","date_gmt":"2025-05-29T17:21:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221488"},"modified":"2025-05-29T17:21:17","modified_gmt":"2025-05-29T17:21:17","slug":"for-the-function-g-whose-graph-is-given-state-the-value-ofeach-quantity-if-it-exists","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/for-the-function-g-whose-graph-is-given-state-the-value-ofeach-quantity-if-it-exists\/","title":{"rendered":"For the function g whose graph is given, state the value ofeach quantity, if it exists"},"content":{"rendered":"\n<p>None<br>For the function g whose graph is given, state the value of<br>each quantity, if it exists. If it does not exist, explain why lim t 2- g(t)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">General Answer<\/h3>\n\n\n\n<p><strong>The left-hand limit of g(t)g(t) as t\u21922\u2212t \\to 2^-:<\/strong> lim\u2061t\u21922\u2212g(t)\\lim_{t \\to 2^-} g(t)<\/p>\n\n\n\n<p>is the value that the function g(t)g(t) approaches <strong>from the left side of 2<\/strong>. You examine the <strong>y-values<\/strong> that g(t)g(t) gets close to as tt gets closer and closer to 2 from values <strong>less than 2<\/strong> (such as 1.9, 1.99, 1.999, etc.).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the graph <strong>smoothly approaches a specific y-value<\/strong> from the left, that value is the left-hand limit.<\/li>\n\n\n\n<li>If the graph <strong>jumps<\/strong> or has a <strong>discontinuity<\/strong> at t=2t = 2, the limit may <strong>still exist from the left<\/strong>, even if the right-hand limit or actual function value at t=2t=2 is different.<\/li>\n\n\n\n<li>If the graph goes off to <strong>infinity<\/strong>, or <strong>oscillates<\/strong>, or has a <strong>vertical asymptote<\/strong>, then the limit <strong>does not exist<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>In calculus, limits help us understand the behavior of a function as its input approaches a certain value. Specifically, the <strong>left-hand limit<\/strong> lim\u2061t\u21922\u2212g(t)\\lim_{t \\to 2^-} g(t) describes how the function behaves as tt approaches 2 from values <strong>less than<\/strong> 2. We only consider values of tt that are slightly less than 2 and observe what y-value g(t)g(t) approaches.<\/p>\n\n\n\n<p>To determine this limit from a graph, we trace the curve of the function as it moves toward t=2t = 2 from the left. If the function approaches a single, definite y-value (say, 5), then the left-hand limit exists and equals that value. This is true even if the function is not defined at t=2t = 2, or if it has a different value at that point. Limits concern only the <strong>approach<\/strong>, not the actual point.<\/p>\n\n\n\n<p>However, if the graph of the function jumps suddenly to another y-value at t=2t = 2, the <strong>left-hand limit may still exist<\/strong>, as long as from the left, the y-values are settling on a specific number. If instead the graph shoots up or down to infinity, or oscillates wildly without settling, then the limit <strong>does not exist<\/strong>.<\/p>\n\n\n\n<p>In short, the limit lim\u2061t\u21922\u2212g(t)\\lim_{t \\to 2^-} g(t) exists <strong>if and only if<\/strong> the graph of g(t)g(t) approaches a <strong>specific, finite y-value<\/strong> from the left. You should use the graph to visually confirm what value, if any, is approached as tt comes from the left.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner8-83.jpeg\" alt=\"\" class=\"wp-image-221489\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>NoneFor the function g whose graph is given, state the value ofeach quantity, if it exists. If it does not exist, explain why lim t 2- g(t) The Correct Answer and Explanation is: General Answer The left-hand limit of g(t)g(t) as t\u21922\u2212t \\to 2^-: lim\u2061t\u21922\u2212g(t)\\lim_{t \\to 2^-} g(t) is the value that the function g(t)g(t) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-221488","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221488","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=221488"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221488\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221488"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221488"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}