{"id":265,"date":"2025-02-13T22:45:46","date_gmt":"2025-02-13T22:45:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-properties-of-limits\/"},"modified":"2025-10-23T19:49:25","modified_gmt":"2025-10-23T19:49:25","slug":"finding-limits-properties-of-limits","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-properties-of-limits\/","title":{"raw":"Properties of Limits: Learn It 1","rendered":"Properties of Limits: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the limit of a sum, a difference, and a product.<\/li>\r\n \t<li style=\"font-weight: 400;\">Find the limit of a polynomial.<\/li>\r\n \t<li style=\"font-weight: 400;\">Find the limit of a power or a root.<\/li>\r\n \t<li style=\"font-weight: 400;\">Find the limit of a quotient.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Limit of a Sum, a Difference, and a Product<\/h2>\r\nGraphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the <strong>properties of limits<\/strong>, which is a collection of theorems for finding limits.\r\n\r\nKnowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Properties of Limits<\/h3>\r\nLet [latex]a,k,A[\/latex], and [latex]B[\/latex] represent real numbers, and [latex]f[\/latex] and [latex]g[\/latex] be functions, such that [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=B[\/latex]. For limits that exist and are finite, the properties of limits are summarized in the table below.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Constant, <em>k<\/em><\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Constant times a function<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[k\\cdot f\\left(x\\right)\\right]=k\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=kA[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sum of functions<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)+\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A+B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Difference of functions<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)-g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)-\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A-B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Product of functions<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\cdot g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)\\cdot \\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A\\cdot B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quotient of functions<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)}{g\\left(x\\right)}=\\dfrac{\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)}{\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)}=\\dfrac{A}{B},B\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Function raised to an exponent<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}{\\left[f\\left(x\\right)\\right]}^{n}={\\left[\\underset{x\\to \\infty }{\\mathrm{lim}}f\\left(x\\right)\\right]}^{n}={A}^{n}[\/latex], where [latex]n[\/latex] is a positive integer<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>n<\/em>th root of a function, where n is a positive integer<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f\\left(x\\right)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\right]}=\\sqrt[n]{A}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Polynomial function<\/td>\r\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}p\\left(x\\right)=p\\left(a\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"159304\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"159304\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)&amp;=\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&amp;&amp; \\text{Sum of functions property} \\\\ &amp;=\\underset{x\\to 3}{2\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&amp;&amp; \\text{Constant times a function property} \\\\ &amp;=2\\left(3\\right)+5 &amp;&amp; \\text{Evaluate} \\\\ &amp;=11 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Evaluate the following limit: [latex]\\underset{x\\to -12}{\\mathrm{lim}}\\left(-2x+2\\right)[\/latex].[reveal-answer q=\"552188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"552188\"]26[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]174089[\/ohm_question]<\/section>\r\n<dl id=\"fs-id1165135453295\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135453300\"><\/dd>\r\n<\/dl>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find the limit of a sum, a difference, and a product.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a polynomial.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a power or a root.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a quotient.<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Limit of a Sum, a Difference, and a Product<\/h2>\n<p>Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the <strong>properties of limits<\/strong>, which is a collection of theorems for finding limits.<\/p>\n<p>Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Properties of Limits<\/h3>\n<p>Let [latex]a,k,A[\/latex], and [latex]B[\/latex] represent real numbers, and [latex]f[\/latex] and [latex]g[\/latex] be functions, such that [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=B[\/latex]. For limits that exist and are finite, the properties of limits are summarized in the table below.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Constant, <em>k<\/em><\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Constant times a function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[k\\cdot f\\left(x\\right)\\right]=k\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=kA[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sum of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)+\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A+B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)-g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)-\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A-B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Product of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\cdot g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)\\cdot \\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A\\cdot B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Quotient of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)}{g\\left(x\\right)}=\\dfrac{\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)}{\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)}=\\dfrac{A}{B},B\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Function raised to an exponent<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}{\\left[f\\left(x\\right)\\right]}^{n}={\\left[\\underset{x\\to \\infty }{\\mathrm{lim}}f\\left(x\\right)\\right]}^{n}={A}^{n}[\/latex], where [latex]n[\/latex] is a positive integer<\/td>\n<\/tr>\n<tr>\n<td><em>n<\/em>th root of a function, where n is a positive integer<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f\\left(x\\right)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\right]}=\\sqrt[n]{A}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Polynomial function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}p\\left(x\\right)=p\\left(a\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q159304\">Show Solution<\/button><\/p>\n<div id=\"q159304\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)&=\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&& \\text{Sum of functions property} \\\\ &=\\underset{x\\to 3}{2\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&& \\text{Constant times a function property} \\\\ &=2\\left(3\\right)+5 && \\text{Evaluate} \\\\ &=11 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Evaluate the following limit: [latex]\\underset{x\\to -12}{\\mathrm{lim}}\\left(-2x+2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q552188\">Show Solution<\/button><\/p>\n<div id=\"q552188\" class=\"hidden-answer\" style=\"display: none\">26<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm174089\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174089&theme=lumen&iframe_resize_id=ohm174089&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<dl 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