{"id":180758,"date":"2025-01-08T20:28:46","date_gmt":"2025-01-08T20:28:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=180758"},"modified":"2025-01-08T20:28:49","modified_gmt":"2025-01-08T20:28:49","slug":"name-1-6-limits-and-continuity-homework-date-period-1","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/08\/name-1-6-limits-and-continuity-homework-date-period-1\/","title":{"rendered":"Name 1.6 Limits and continuity Homework Date Period 1"},"content":{"rendered":"\n

Name 1.6 Limits and continuity Homework Date Period 1. Use the graph of the function y = g(x) shown below, to evaluate each of the following. A. lim g(x) = B. lim 9(x) = ya Clim, g(x) = D. 9(-1) = ad E. lim g(x) = Elim, g(x) = . Glim g(x) = H. lim g(x) = J. g(7) = K. lim g(x) = 2. Use the graph of the function y = f(x) shown<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To answer your homework questions, let’s first define the basic concepts of limits and continuity in calculus:<\/p>\n\n\n\n

1. Limit Definition<\/strong>:<\/h3>\n\n\n\n

The limit of a function f(x)f(x) as xx approaches a value aa is the value that f(x)f(x) gets closer to as xx gets closer to aa. In mathematical notation: lim\u2061x\u2192af(x)=L\\lim_{{x \\to a}} f(x) = L<\/p>\n\n\n\n

if for every small value \u03f5>0\\epsilon > 0, there is a small value \u03b4>0\\delta > 0 such that for all xx within the range (a\u2212\u03b4,a+\u03b4)(a – \\delta, a + \\delta), f(x)f(x) stays within the range (L\u2212\u03f5,L+\u03f5)(L – \\epsilon, L + \\epsilon).<\/p>\n\n\n\n

2. Continuity<\/strong>:<\/h3>\n\n\n\n

A function f(x)f(x) is continuous at a point aa if:<\/p>\n\n\n\n

    \n
  • f(a)f(a) exists.<\/li>\n\n\n\n
  • The limit lim\u2061x\u2192af(x)\\lim_{{x \\to a}} f(x) exists.<\/li>\n\n\n\n
  • lim\u2061x\u2192af(x)=f(a)\\lim_{{x \\to a}} f(x) = f(a).<\/li>\n<\/ul>\n\n\n\n

    Now let’s proceed with the evaluation of limits and function values using the graph of g(x)g(x) and f(x)f(x). I would need to see the graph of these functions in order to give precise answers. However, I can still provide a general approach for how you can evaluate these limits from the graph.<\/h3>\n\n\n\n
    \n\n\n\n

    Example Evaluation Based on the Graph:<\/h3>\n\n\n\n
      \n
    1. A. lim\u2061x\u2192ag(x)\\lim_{{x \\to a}} g(x)<\/strong>: Look at the graph near x=ax = a. If the graph approaches a particular value from both sides of x=ax = a, then that value is the limit of g(x)g(x) as x\u2192ax \\to a.<\/li>\n\n\n\n
    2. B. lim\u2061x\u2192a+g(x)\\lim_{{x \\to a^+}} g(x)<\/strong>: This is the limit of g(x)g(x) as xx approaches aa from the right (from values larger than aa). Look at the graph from the right side of x=ax = a to find this limit.<\/li>\n\n\n\n
    3. C. lim\u2061x\u2192a\u2212g(x)\\lim_{{x \\to a^-}} g(x)<\/strong>: This is the limit of g(x)g(x) as xx approaches aa from the left (from values smaller than aa). Again, look at the graph from the left side of x=ax = a to evaluate this limit.<\/li>\n\n\n\n
    4. D. g(\u22121)g(-1)<\/strong>: To evaluate g(\u22121)g(-1), simply locate the point on the graph where x=\u22121x = -1. The corresponding value of yy at x=\u22121x = -1 is g(\u22121)g(-1).<\/li>\n\n\n\n
    5. E. lim\u2061x\u2192bg(x)\\lim_{{x \\to b}} g(x)<\/strong>: Similar to the first step, but look at x=bx = b this time. The limit will be the value the function is approaching from both sides.<\/li>\n\n\n\n
    6. F. lim\u2061x\u2192b+g(x)\\lim_{{x \\to b^+}} g(x)<\/strong> and G. lim\u2061x\u2192b\u2212g(x)\\lim_{{x \\to b^-}} g(x)<\/strong>: These are the right-hand and left-hand limits, respectively. Use the graph to analyze the function’s behavior from the right and left of x=bx = b.<\/li>\n\n\n\n
    7. H. lim\u2061x\u2192cg(x)\\lim_{{x \\to c}} g(x)<\/strong>: Look at the graph near x=cx = c. If the function is approaching a specific value from both directions, that\u2019s your answer.<\/li>\n\n\n\n
    8. I. g(7)g(7)<\/strong>: Find the value of the function at x=7x = 7. Look for the point where x=7x = 7 on the graph and note the corresponding value of g(7)g(7).<\/li>\n\n\n\n
    9. J. lim\u2061x\u2192dg(x)\\lim_{{x \\to d}} g(x)<\/strong>: Evaluate the limit at x=dx = d similarly to previous steps. If the graph shows a jump or discontinuity at dd, consider if the left-hand limit and right-hand limit are equal.<\/li>\n\n\n\n
    10. K. lim\u2061x\u2192eg(x)\\lim_{{x \\to e}} g(x)<\/strong>: Evaluate the limit at x=ex = e. If the graph approaches a specific value from both directions, that\u2019s your limit.<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      Explanation:<\/h3>\n\n\n\n
        \n
      • Continuity<\/strong>: If the left-hand limit, right-hand limit, and the function’s value at a specific point are all equal, then the function is continuous at that point.<\/li>\n\n\n\n
      • Discontinuities<\/strong>: If there is a jump or the left-hand and right-hand limits do not match, the function is discontinuous at that point.<\/li>\n<\/ul>\n\n\n\n

        These general steps should help you evaluate limits and function values directly from the graph.<\/p>\n\n\n\n

        If you can provide the graph or describe it in more detail, I can give you specific answers based on the given functions.<\/p>\n","protected":false},"excerpt":{"rendered":"

        Name 1.6 Limits and continuity Homework Date Period 1. Use the graph of the function y = g(x) shown below, to evaluate each of the following. A. lim g(x) = B. lim 9(x) = ya Clim, g(x) = D. 9(-1) = ad E. lim g(x) = Elim, g(x) = . Glim g(x) = H. lim […]<\/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-180758","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180758","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=180758"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180758\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=180758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=180758"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=180758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}