{"id":27895,"date":"2024-08-15T19:51:17","date_gmt":"2024-08-15T19:51:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27895"},"modified":"2024-08-15T19:51:17","modified_gmt":"2024-08-15T19:51:17","slug":"solved-problems-finite-limits","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/solved-problems-finite-limits\/","title":{"rendered":"Solved Problems: Finite Limits"},"content":{"rendered":"<p><center><\/p>\n<h1>Solved Problems: Finite Limits<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nThis class offers solved problems on the calculation of finite limits, accompanied by a video tutorial. Students are guided to solve the problems and compare their results with the provided solutions. Key techniques such as rationalization and the use of graphs are highlighted, encouraging different approaches to reach the correct answer.<\/em><\/p>\n<p><strong>Learning Objectives<\/strong><br \/>\nBy the end of this class, the student will be able to:<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Calculate<\/strong> finite limits of polynomial and rational functions.<\/li>\n<li><strong>Use<\/strong> rationalization techniques to simplify limit calculations.<\/li>\n<li><strong>Compare<\/strong> different solving methods and recognize the existence of multiple paths to the correct solution.<\/li>\n<\/ul>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/VY1gr1EUq3o\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">These solved exercises will help you master the calculation of finite limits. The solutions are in the video.<\/p>\n<p style=\"text-align: justify; color: #000000;\">To enhance the learning experience, I have created this video with solved exercises on limit calculation, showing all the particularities that can be inferred from them.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Try to solve these exercises on your own and then compare your results.<br \/>\nNext to each exercise, you will find a link that leads to the part of the YouTube video where all the calculations are performed up to the solution.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Sometimes you will have to rationalize, and other times you will have to rely on the definition of a limit to justify an answer. Sometimes, even a graph can save you a lot of work when deciding which line of reasoning to follow.<\/p>\n<h3>I. Calculate the following finite limits:<\/h3>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -1} (x^3+2x^2 - 3x - 4) <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=35s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0} \\dfrac{3^x - 3^{-x}}{3^x + 3^{-x}} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=64s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2} \\dfrac{x-1}{x^2-1} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=91s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2} \\dfrac{x^2-4}{x^2-5x+6} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=161s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -1} \\dfrac{x^2+3x+2}{x^2+4x+3} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=351s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2} \\dfrac{x-2}{\\sqrt{x^2-4}} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=594s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2} \\dfrac{\\sqrt{x-2}}{x^2-4} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=887s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ol>\n<h3>II. Find <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{h\\to 0} \\frac{f(x+h)-f(x)}{h}<\/span><\/span> for the following functions:<\/h3>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\dfrac{1}{x-2} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=1088s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\sqrt{x-4} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=1263s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\dfrac{x}{x+1} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=VY1gr1EUq3o&amp;t=1366s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">Have you managed to solve them all on your own?<\/p>\n<p style=\"text-align: justify; color: #000000;\">How different are your solutions from mine?<\/p>\n<p style=\"text-align: justify; color: #000000;\">Remember that the path to solving a problem is not necessarily unique.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solved Problems: Finite Limits Summary: This class offers solved problems on the calculation of finite limits, accompanied by a video tutorial. Students are guided to solve the problems and compare their results with the provided solutions. Key techniques such as rationalization and the use of graphs are highlighted, encouraging different approaches to reach the correct [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27893,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[854,567],"tags":[],"class_list":["post-27895","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-differential-calculus","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Solved Problems: Finite Limits - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Master the calculation of finite limits with this class that includes solved problems. 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