{"id":251670,"date":"2025-07-10T19:22:36","date_gmt":"2025-07-10T19:22:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=251670"},"modified":"2025-07-10T19:22:39","modified_gmt":"2025-07-10T19:22:39","slug":"lim-h-0-fx-8-ixl-choose-the-correct-answer-below","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/lim-h-0-fx-8-ixl-choose-the-correct-answer-below\/","title":{"rendered":"lim h-0 f(x) = 8 &#8211; Ixl Choose the correct answer below"},"content":{"rendered":"\n<p>lim h-0 f(x) = 8 &#8211; Ixl Choose the correct answer below: The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient exist at x = 0, but are not equal: The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient do not exist at x= 0. The function is differentiable at x = 0 because the graph has a sharp corner atx= 0. The function is differentiable at x = 0 because both the left- and right-hand limits of the difference quotient exist at x = 0.<br>f(0 + h) &#8211; f(0) Determine whether is differentiable at x = 0 by considering lim h-0 f(x) = 8 &#8211; Ixl Choose the correct answer below: The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient exist at x = 0, but are not equal: The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient do not exist at x= 0. The function is differentiable at x = 0 because the graph has a sharp corner atx= 0. The function is differentiable at x = 0 because both the left- and right-hand limits of the difference quotient exist at x = 0.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The function in question is f(x)=8\u2212\u2223x\u2223f(x) = 8 &#8211; |x|f(x)=8\u2212\u2223x\u2223, and we are interested in determining its differentiability at x=0x = 0x=0.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Calculate the left-hand and right-hand limits of the difference quotient.<\/h3>\n\n\n\n<p>To determine whether the function is differentiable at x=0x = 0x=0, we need to check the behavior of the difference quotient as h\u21920h \\to 0h\u21920. The difference quotient is:f(x+h)\u2212f(x)h\\frac{f(x + h) &#8211; f(x)}{h}hf(x+h)\u2212f(x)\u200b<\/p>\n\n\n\n<p>At x=0x = 0x=0, this becomes:f(0+h)\u2212f(0)h=f(h)\u2212f(0)h\\frac{f(0 + h) &#8211; f(0)}{h} = \\frac{f(h) &#8211; f(0)}{h}hf(0+h)\u2212f(0)\u200b=hf(h)\u2212f(0)\u200b<\/p>\n\n\n\n<p>We know that f(x)=8\u2212\u2223x\u2223f(x) = 8 &#8211; |x|f(x)=8\u2212\u2223x\u2223, so:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>When x>0x > 0x>0, f(x)=8\u2212xf(x) = 8 &#8211; xf(x)=8\u2212x.<\/li>\n\n\n\n<li>When x&lt;0x &lt; 0x&lt;0, f(x)=8+xf(x) = 8 + xf(x)=8+x.<\/li>\n\n\n\n<li>f(0)=8f(0) = 8f(0)=8 (since f(0)=8\u2212\u22230\u2223=8f(0) = 8 &#8211; |0| = 8f(0)=8\u2212\u22230\u2223=8).<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">For h&gt;0h &gt; 0h&gt;0 (right-hand limit):<\/h4>\n\n\n\n<p>f(h)\u2212f(0)h=(8\u2212h)\u22128h=\u2212hh=\u22121\\frac{f(h) &#8211; f(0)}{h} = \\frac{(8 &#8211; h) &#8211; 8}{h} = \\frac{-h}{h} = -1hf(h)\u2212f(0)\u200b=h(8\u2212h)\u22128\u200b=h\u2212h\u200b=\u22121<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">For h&lt;0h &lt; 0h&lt;0 (left-hand limit):<\/h4>\n\n\n\n<p>f(h)\u2212f(0)h=(8+h)\u22128h=hh=1\\frac{f(h) &#8211; f(0)}{h} = \\frac{(8 + h) &#8211; 8}{h} = \\frac{h}{h} = 1hf(h)\u2212f(0)\u200b=h(8+h)\u22128\u200b=hh\u200b=1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Evaluate the limits<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>As h\u21920+h \\to 0^+h\u21920+ (from the right), the limit of the difference quotient is \u22121-1\u22121.<\/li>\n\n\n\n<li>As h\u21920\u2212h \\to 0^-h\u21920\u2212 (from the left), the limit of the difference quotient is 111.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Conclusion<\/h3>\n\n\n\n<p>Since the left-hand and right-hand limits of the difference quotient are not equal, the function is <strong>not differentiable at x=0x = 0x=0<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Correct Answer:<\/h3>\n\n\n\n<p><strong>The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient exist at x = 0, but are not equal.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The fact that the left-hand and right-hand limits are different (one is \u22121-1\u22121 and the other is 111) means that the slope of the function from the left side is different from the slope from the right side at x=0x = 0x=0. This is a clear indicator of a &#8220;corner&#8221; in the graph of the function at that point, which causes the function to fail the differentiability test at x=0x = 0x=0. Therefore, the function is not differentiable at this point, and this sharp corner behavior is a key feature of functions like f(x)=8\u2212\u2223x\u2223f(x) = 8 &#8211; |x|f(x)=8\u2212\u2223x\u2223.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-174.jpeg\" alt=\"\" class=\"wp-image-251682\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>lim h-0 f(x) = 8 &#8211; Ixl Choose the correct answer below: The function is not differentiable at x = 0 because the left- and right-hand limits of the difference quotient exist at x = 0, but are not equal: The function is not differentiable at x = 0 because the left- and right-hand limits [&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-251670","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251670","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=251670"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251670\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=251670"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=251670"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=251670"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}