{"id":188473,"date":"2025-02-07T04:19:19","date_gmt":"2025-02-07T04:19:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188473"},"modified":"2025-02-07T04:19:21","modified_gmt":"2025-02-07T04:19:21","slug":"from-ap-live-calculus-ab-april-13-unit-1-limits-and-continuity","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/07\/from-ap-live-calculus-ab-april-13-unit-1-limits-and-continuity\/","title":{"rendered":"From AP Live Calculus AB – April 13; Unit 1: Limits And Continuity"},"content":{"rendered":"\n

From AP Live Calculus AB – April 13; Unit 1: Limits And Continuity 1-3: Use The Graph Of The Piecewise Function, F, Shown To Find The Requested Values. 1. F(0) En 2. Lim F(X) (3. 1 3. Lim F(X) X – 3 4. X + 5 X=-5 X2 – 25 Lim – 2x +3 – 3 5. Lim X+ 3x – 3<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To evaluate the given limits and function values, we need to analyze the piecewise function ( f(x) ) based on its graph. Here’s how to approach each part:<\/p>\n\n\n\n

    \n
  1. ( f(0) ):<\/strong> – ocate ( x = 0 ) on the graph. – dentify the value of ( f(x) ) at this point. – f the graph shows a filled circle at ( x = 0 ), then ( f(0) ) is the value at that point. – f there’s an open circle, ( f(0) ) is undefined.<\/li>\n\n\n\n
  2. ( \\lim_{x \\to 3} f(x) ):<\/strong> – xamine the behavior of ( f(x) ) as ( x ) approaches 3 from both the left and right. – f the function approaches the same value from both sides, that value is the limit. – f the function approaches different values from each side, the limit does not exist.<\/li>\n\n\n\n
  3. ( \\lim_{x \\to 3^-} f(x) ):<\/strong> – ocus on the left-hand behavior of ( f(x) ) as ( x ) approaches 3. – dentify the value the function approaches from the left side.<\/li>\n\n\n\n
  4. ( \\lim_{x \\to -5} \\frac{x + 5}{x^2 – 25} ):<\/strong> – actor the denominator: ( x^2 – 25 = (x + 5)(x – 5) ). – implify the expression: ( \\frac{x + 5}{(x + 5)(x – 5)} = \\frac{1}{x – 5} ). – s ( x ) approaches -5, the expression becomes ( \\frac{1}{-5 – 5} = \\frac{1}{-10} ). – herefore, ( \\lim_{x \\to -5} \\frac{x + 5}{x^2 – 25} = -\\frac{1}{10} ).<\/li>\n\n\n\n
  5. ( \\lim_{x \\to 3} \\frac{\\sqrt{2x + 3} – 3}{x – 3} ):<\/strong> – irect substitution leads to the indeterminate form ( \\frac{0}{0} ). – ultiply the numerator and denominator by the conjugate of the numerator: [ \\frac{\\sqrt{2x + 3} – 3}{x – 3} \\times \\frac{\\sqrt{2x + 3} + 3}{\\sqrt{2x + 3} + 3} = \\frac{(2x + 3) – 9}{(x – 3)(\\sqrt{2x + 3} + 3)} = \\frac{2x – 6}{(x – 3)(\\sqrt{2x + 3} + 3)} ] – actor the numerator: [ \\frac{2(x – 3)}{(x – 3)(\\sqrt{2x + 3} + 3)} ] – ancel the ( (x – 3) ) terms: [ \\frac{2}{\\sqrt{2x + 3} + 3} ] – ubstitute ( x = 3 ): [ \\frac{2}{\\sqrt{2(3) + 3} + 3} = \\frac{2}{\\sqrt{9} + 3} = \\frac{2}{3 + 3} = \\frac{2}{6} = \\frac{1}{3} ] – herefore, ( \\lim_{x \\to 3} \\frac{\\sqrt{2x + 3} – 3}{x – 3} = \\frac{1}{3} ).
    For a more in-depth understanding of limits and continuity, you might find the following video helpful:<\/li>\n<\/ol>\n\n\n\n

    \ue200video\ue2022022 Live Review 3 | AP Calculus AB | How to Find Limits & Continuity\ue202turn0search2\ue201<\/p>\n","protected":false},"excerpt":{"rendered":"

    From AP Live Calculus AB – April 13; Unit 1: Limits And Continuity 1-3: Use The Graph Of The Piecewise Function, F, Shown To Find The Requested Values. 1. F(0) En 2. Lim F(X) (3. 1 3. Lim F(X) X – 3 4. X + 5 X=-5 X2 – 25 Lim – 2x +3 – […]<\/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-188473","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188473","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=188473"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188473\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}