{"id":1610,"date":"2025-07-25T04:06:35","date_gmt":"2025-07-25T04:06:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1610"},"modified":"2026-02-13T17:45:40","modified_gmt":"2026-02-13T17:45:40","slug":"properties-of-limits-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/properties-of-limits-fresh-take\/","title":{"raw":"Properties of Limits: Fresh Take","rendered":"Properties of Limits: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the limit of a polynomial.<\/li>\r\n \t<li>Find the limit of a power or a root.<\/li>\r\n \t<li>Find the limit of a quotient.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluating Limits<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Limit Laws:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Sum\/Difference: [latex]\\lim_{x \\to a} [f(x) \\pm g(x)] = \\lim_{x \\to a} f(x) \\pm \\lim_{x \\to a} g(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Product: [latex]\\lim_{x \\to a} [f(x) \\cdot g(x)] = \\lim_{x \\to a} f(x) \\cdot \\lim_{x \\to a} g(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quotient: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{\\lim_{x \\to a} f(x)}{\\lim_{x \\to a} g(x)}[\/latex], if [latex]\\lim_{x \\to a} g(x) \\neq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Power: [latex]\\lim_{x \\to a} [f(x)]^n = [\\lim_{x \\to a} f(x)]^n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Polynomial and Rational Function Limits:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} p(x) = p(a)[\/latex] for polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} \\frac{p(x)}{q(x)} = \\frac{p(a)}{q(a)}[\/latex] for rational functions, if [latex]q(a) \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Techniques for Indeterminate Forms:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Factoring and Simplifying:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Factor polynomials and cancel common terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rationalizing (Multiplying by Conjugate):\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Used for limits with square roots<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplifying Complex Fractions:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Combine fractions using LCD<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Indeterminate Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{0}{0}[\/latex], [latex]\\frac{\\infty}{\\infty}[\/latex], [latex]0 \\cdot \\infty[\/latex], etc.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Require special techniques for evaluation<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\r\n[reveal-answer q=\"6635113\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6635113\"]\r\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572094142\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572094142\"]\r\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\r\n[reveal-answer q=\"4482011\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4482011\"]\r\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571688072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571688072\"]\r\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571598007\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{x^2+4x+3}{x^2-9}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571598067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571598067\"]\r\n<p id=\"fs-id1170571598067\">[latex]\\dfrac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571611956\">Evaluate [latex]\\underset{x\\to 5}{\\lim}\\dfrac{\\sqrt{x-1}-2}{x-5}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571612008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571612008\"]\r\n<p id=\"fs-id1170571612008\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170572394360\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{\\frac{1}{x+2}+1}{x+3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571648126\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571648126\"]\r\n<p id=\"fs-id1170571648126\">[latex]\u22121[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571681429\">Evaluate [latex]\\underset{x\\to 3}{\\lim}\\left(\\dfrac{1}{x-3}-\\dfrac{4}{x^2-2x-3}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"80944622\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"80944622\"]\r\n<p id=\"fs-id1170572233797\">Don\u2019t forget to factor [latex]x^2-2x-3[\/latex] before getting a common denominator.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572233826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572233826\"]\r\n<p id=\"fs-id1170572233826\">[latex]\\frac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the limit of a polynomial.<\/li>\n<li>Find the limit of a power or a root.<\/li>\n<li>Find the limit of a quotient.<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluating Limits<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Limit Laws:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum\/Difference: [latex]\\lim_{x \\to a} [f(x) \\pm g(x)] = \\lim_{x \\to a} f(x) \\pm \\lim_{x \\to a} g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product: [latex]\\lim_{x \\to a} [f(x) \\cdot g(x)] = \\lim_{x \\to a} f(x) \\cdot \\lim_{x \\to a} g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{\\lim_{x \\to a} f(x)}{\\lim_{x \\to a} g(x)}[\/latex], if [latex]\\lim_{x \\to a} g(x) \\neq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power: [latex]\\lim_{x \\to a} [f(x)]^n = [\\lim_{x \\to a} f(x)]^n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial and Rational Function Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} p(x) = p(a)[\/latex] for polynomials<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} \\frac{p(x)}{q(x)} = \\frac{p(a)}{q(a)}[\/latex] for rational functions, if [latex]q(a) \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Techniques for Indeterminate Forms:\n<ul>\n<li class=\"whitespace-normal break-words\">Factoring and Simplifying:\n<ul>\n<li class=\"whitespace-normal break-words\">Factor polynomials and cancel common terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rationalizing (Multiplying by Conjugate):\n<ul>\n<li class=\"whitespace-normal break-words\">Used for limits with square roots<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplifying Complex Fractions:\n<ul>\n<li class=\"whitespace-normal break-words\">Combine fractions using LCD<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Indeterminate Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{0}{0}[\/latex], [latex]\\frac{\\infty}{\\infty}[\/latex], [latex]0 \\cdot \\infty[\/latex], etc.<\/li>\n<li class=\"whitespace-normal break-words\">Require special techniques for evaluation<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6635113\">Hint<\/button><\/p>\n<div id=\"q6635113\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572094142\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572094142\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4482011\">Hint<\/button><\/p>\n<div id=\"q4482011\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571688072\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571688072\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571598007\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{x^2+4x+3}{x^2-9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571598067\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571598067\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571598067\">[latex]\\dfrac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571611956\">Evaluate [latex]\\underset{x\\to 5}{\\lim}\\dfrac{\\sqrt{x-1}-2}{x-5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571612008\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571612008\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612008\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572394360\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{\\frac{1}{x+2}+1}{x+3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571648126\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571648126\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571648126\">[latex]\u22121[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571681429\">Evaluate [latex]\\underset{x\\to 3}{\\lim}\\left(\\dfrac{1}{x-3}-\\dfrac{4}{x^2-2x-3}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q80944622\">Hint<\/button><\/p>\n<div id=\"q80944622\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572233797\">Don\u2019t forget to factor [latex]x^2-2x-3[\/latex] before getting a common denominator.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572233826\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572233826\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572233826\">[latex]\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1610"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1610\/revisions"}],"predecessor-version":[{"id":5644,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1610\/revisions\/5644"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1610\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1610"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1610"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1610"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1610"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}