{"id":15702,"date":"2022-12-19T07:00:17","date_gmt":"2022-12-19T07:00:17","guid":{"rendered":"https:\/\/statanalytica.com\/blog\/?p=15702"},"modified":"2023-09-22T07:10:49","modified_gmt":"2023-09-22T06:10:49","slug":"types-and-rules-of-limit-calculus","status":"publish","type":"post","link":"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/","title":{"rendered":"What Are The Types And Rules Of Limit Calculus?"},"content":{"rendered":"\n<p>The concept of limit calculus is one of the most essential concepts. Calculus is one of the most important concepts in mathematics, and the limit is one of its most fundamental concepts. Without it, <a href=\"https:\/\/statanalytica.com\/blog\/basic-calculus-formula\/\" target=\"_blank\" rel=\"noreferrer noopener\">calculus<\/a> would be much more difficult to understand.<\/p>\n\n\n\n<p>In this article, we\u2019ll provide a brief overview of the limit and outline the types of limits that exist: one-sided, two-sided, etc. We will also provide instructions on how to calculate limits using the rules given.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"what-is-the-limit\"><\/span><strong>What is the limit?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69f333d97cbe3\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #ff5104;color:#ff5104\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #ff5104;color:#ff5104\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69f333d97cbe3\" checked aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#what-is-the-limit\" >What is the limit?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#types-of-limits-in-calculus\" >Types of limits in calculus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#rules-of-limit-calculus\" >Rules of limit calculus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#some-well-known-formulas-of-limit-calculus\" >Some well-known formulas of limit calculus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#how-to-calculate-the-limit-calculus-problems\" >How to calculate the limit calculus problems?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/statanalytica.com\/blog\/types-and-rules-of-limit-calculus\/#sum-up\" >Sum Up<\/a><\/li><\/ul><\/nav><\/div>\n\n\n\n\n<p>In mathematical terms, a limit is a point at which a function reaches its maximum or minimum value. A limit is simply something that you cannot exceed. It&#8217;s used to find out exactly how a function will change as the input values get closer and closer to some given value.\u00a0<\/p>\n\n\n\n<p>This can be helpful in many different circumstances, such as when trying to figure out how much money someone will make over their lifetime, or when trying to design a machine that can handle more complicated tasks.<\/p>\n\n\n\n<p>The general expression of limit calculus is:<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> f(u) = M<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"types-of-limits-in-calculus\"><\/span><strong>Types of limits in calculus<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>There are different types of limits:&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Pointwise limits<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Pointwise limits are the simplest type and they are just what they sound like &#8211; the limit is found at one specific point in space. The function of the pointwise limit will be:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/y16VO7mKdpGZaBuDjh_-0vL9cMIhrcemzmJkCsDFowxguA06b-arDwFrl8_dy891Flwm7iHLBg8rr1ANKU1L89wP-c3390RMymLKiBuFsOeMOC9EMv-GtReFfyKdTUaRczuw3Dw_XZNSEd52qzRhNtOa8IY-iMktqtbFOTuJrc85IbHz-5LZUVIhC4MMVjivHMosLZe2wg\" alt=\"\"\/><\/figure>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li><strong>One-sided limits<\/strong><\/li>\n<\/ol>\n\n\n\n<p>One-sided limits involve finding the limit as both the input values and output values approach a single point in space. It can be either a left-hand limit or a right-hand limit.<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong><sup>+<\/sup><\/strong><strong> f(u) = M<\/strong><\/p>\n\n\n\n<p><strong>Or<\/strong><\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong><sup>&#8211;<\/sup><\/strong><strong> f(u) = M<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li><strong>Two-sided limits<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Two-sided limits involve finding both the pointwise and one-sided limits at different points in space. When both left-hand and right-hand limits exist, we will get the two-sided limits.<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> f(u) = M<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li><strong>Limits at infinity<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Limits at infinity are important because they tell us how far away from a given point an object or function can get before it stops changing or reaching a new level.<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192<\/sub><\/strong><sub>\u221e<\/sub><strong> f(u) = M&nbsp;<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"rules-of-limit-calculus\"><\/span><strong>Rules of limit calculus<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Here are some well-known rules of limit calculus.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Constant Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The constant rule of limit calculus is used when a function is given in which there is no independent variable available. This rule state that the limit of the constant function remains unchanged.<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> k = k<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li><strong>Constant time function rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>This rule is used when there are constant coefficients along with the independent variables of the function. This rule states that the constant coefficient will be written outside the limit notation.<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> kf(u) = k Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> f(u)<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li><strong>Sum Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The sum rule is used when there are two or more terms or functions given along with the plus sign among them. This rule states that the notation of the limit will apply to each function separately.&nbsp;<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u) + g(u)] = Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u)] + Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [g(u)]<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li><strong>Difference Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The difference rule is used when there are two or more terms or functions are given along with the minus sign among them. This rule states that the notation of the limit will apply to each function separately.&nbsp;<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u) &#8211; g(u)] = Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u)] &#8211; Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [g(u)]<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"5\">\n<li><strong>Product Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The product rule is used when there are two or more terms or functions given along with the multiply sign among them. This rule states that the notation of the limit will apply to each function separately.&nbsp;<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u) x g(u)] = Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u)] x Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [g(u)]<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"6\">\n<li><strong>Quotient Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The quotient rule is used when there are two or more terms or functions given along with the division sign among them. This rule states that the notation of the limit will apply to each function separately.&nbsp;<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u) \/ g(u)] = Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u)] \/ Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [g(u)]<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"7\">\n<li><strong>L\u2019hopital\u2019s Rule<\/strong><\/li>\n<\/ol>\n\n\n\n<p>L\u2019hopital\u2019s rule is used when a function makes an undefined form such as 0\/0, \u221e\/\u221e, \u221e<sup>\u221e<\/sup>, etc. This rule states that if the function forms an undefined form, then take the derivative of the given function with respect to the corresponding variable.<\/p>\n\n\n\n<p>After that apply the limit value again. If the function again forms an undefined form then take the second derivative of the function, and so on until you get the result that is defined.&nbsp;<\/p>\n\n\n\n<p><strong>Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [f(u) \/ g(u)] = Lim<\/strong><strong><sub>u\u2192a<\/sub><\/strong><strong> [d\/du f(u) \/ d\/du g(u)]<\/strong><\/p>\n\n\n\n<p>To get the step-by-step solution to complex limit problems according to the above rules, try a limit calculator with steps by Allmath.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/LFq_WTaRBuKOF09cfMz_T7FOzYoDNl1ONp_yTyHf_zJDaX4SBgobIPRyiPHnFrLc78rEetGYpxX97h9qPW1Tzd5AVRXE6MRSVCA4Jp09qmClOmqF-80zh_E9g6ZzqiQSeH3MYGZeUttpnP4HagE83gRjBgHCPFsxLNNe4RNElqPbzaOuGRq-4b-vilONm_5ffmVMtn3qzg\" alt=\"limit calculator\n\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"some-well-known-formulas-of-limit-calculus\"><\/span><strong>Some well-known formulas of limit calculus<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Here is a list of some well-known formulas of limit in calculus.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Formula 1<\/td><td>Lim<sub>u\u21920<\/sub> e<sup>u<\/sup> = 1<\/td><\/tr><tr><td>Formula 2<\/td><td>Lim<sub>u\u21920<\/sub> [(e<sup>u<\/sup> \u2013 1)\/u] = 1<\/td><\/tr><tr><td>Formula 3<\/td><td>Lim<sub>u\u21920<\/sub> [sin(u)\/u] = 1<\/td><\/tr><tr><td>Formula 4<\/td><td>Lim<sub>u\u21920<\/sub> [log(1 + u)\/u] = 1<\/td><\/tr><tr><td>Formula 5<\/td><td>Lim<sub>u\u21920<\/sub> [(a<sup>u<\/sup> \u2013 1)\/u] = log<sub>e<\/sub> a<\/td><\/tr><tr><td>Formula 6<\/td><td>Lim<sub>u\u21920<\/sub> [1 + u]<sup>1\/u<\/sup> = e<\/td><\/tr><tr><td>Formula 7<\/td><td>Lim<sub>u\u21920<\/sub> [1 + 1\/u]<sup>u<\/sup> = e<\/td><\/tr><tr><td>Formula 8<\/td><td>Lim<sub>u\u21920<\/sub> [1 + a\/u]<sup>u<\/sup> = e<sup>a<\/sup><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"how-to-calculate-the-limit-calculus-problems\"><\/span><strong>How to calculate the limit calculus problems?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>The problems of limit calculus can be calculated easily with the help of limit rules. Let us take a few examples to understand it.<\/p>\n\n\n\n<p><strong>Example 1<\/strong><\/p>\n\n\n\n<p>Evaluate the limit of g(w) = 2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>, as the specific point is 3.<\/p>\n\n\n\n<p><strong>Solution&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Step 1:<\/strong> First of all, write the given function in the form general limit expression.<\/p>\n\n\n\n<p>Lim<sub>w\u2192a<\/sub> [g(w)] = Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>]<\/p>\n\n\n\n<p><strong>Step 2: <\/strong>Now apply the sum, difference, and product rules of limit calculus as there is a plus, minus, and multiply sign among the terms of the function.&nbsp;<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup>] + Lim<sub>w\u21923<\/sub> [12w<sup>4<\/sup>] &#8211; Lim<sub>w\u21923<\/sub> [6w<sup>6<\/sup>] * Lim<sub>w\u21923<\/sub> [8w<sup>2<\/sup>]<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Now use the constant times function rules as there are constant coefficients along with the independent variable.<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 2Lim<sub>w\u21923<\/sub> [w<sup>2<\/sup>] + 12Lim<sub>w\u21923<\/sub> [w<sup>4<\/sup>] &#8211; 6Lim<sub>w\u21923<\/sub> [w<sup>6<\/sup>] * 8Lim<sub>w\u21923<\/sub> [w<sup>2<\/sup>]<\/p>\n\n\n\n<p><strong>Step 4: <\/strong>Now apply the specific point to the above expression in place of the independent variable.&nbsp;<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 2 [3<sup>2<\/sup>] + 12 [3<sup>4<\/sup>] &#8211; 6 [3<sup>6<\/sup>] * 8 [3<sup>2<\/sup>]<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 2 [9] + 12 [81] &#8211; 6 [729] * 8 [9]<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 18 + 972 &#8211; 4374 * 72<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 18 + 972 &#8211; 314928<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = 990 &#8211; 314928<\/p>\n\n\n\n<p>Lim<sub>w\u21923<\/sub> [2w<sup>2<\/sup> + 12w<sup>4<\/sup> &#8211; 6w<sup>6<\/sup> * 8w<sup>2<\/sup>] = -313938<\/p>\n\n\n\n<p><strong>Example 2<\/strong><\/p>\n\n\n\n<p>Evaluate the limit of g(x) = (2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3) as the specific point is 1.<\/p>\n\n\n\n<p><strong>Solution&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Step 1:<\/strong> First of all, write the given function in the form general limit expression.<\/p>\n\n\n\n<p>Lim<sub>x\u2192a<\/sub> [g(x)] = Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)]<\/p>\n\n\n\n<p><strong>Step 2: <\/strong>Now apply the difference and quotient rules of limit calculus as there is minus and division sign among the terms of the function.&nbsp;<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (Lim<sub>x\u21921<\/sub> [2x<sup>2<\/sup>] \u2013 Lim<sub>x\u21921<\/sub> [x] \u2013 Lim<sub>x\u21921<\/sub> [1]) \/ (Lim<sub>x\u21921<\/sub> [3x] \u2013 Lim<sub>x\u21921<\/sub> [3])<\/p>\n\n\n\n<p><strong>Step 3:<\/strong> Now use the constant times function rules as there are constant coefficients along with the independent variable.<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (2Lim<sub>x\u21921<\/sub> [x<sup>2<\/sup>] \u2013 Lim<sub>x\u21921<\/sub> [x] \u2013 Lim<sub>x\u21921<\/sub> [1]) \/ (3Lim<sub>x\u21921<\/sub> [x] \u2013 Lim<sub>x\u21921<\/sub> [3])<\/p>\n\n\n\n<p><strong>Step 4: <\/strong>Now apply the specific point to the above expression in the place of the independent variable.&nbsp;<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (2[1<sup>2<\/sup>] \u2013 [1] \u2013 [1]) \/ (3 [1] \u2013 [3])<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (2[1] \u2013 [1] \u2013 [1]) \/ (3 [1] \u2013 [3])<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (2 \u2013 1 \u2013 1) \/ (3 \u2013 3)<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = (2 \u2013 2) \/ (3 \u2013 3)<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = 0\/0<\/p>\n\n\n\n<p><strong>Step 5:<\/strong> Now use L\u2019hopital\u2019s law of limit calculus as the given function makes an undefined form.<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = Lim<sub>x\u21921<\/sub> [d\/dx (2x<sup>2<\/sup> \u2013 x \u2013 1) \/ d\/dx (3x \u2013 3)]<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = Lim<sub>x\u21921<\/sub> [(4x \u2013 1 \u2013 0) \/ (3 \u2013 0)]<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = Lim<sub>x\u21921<\/sub> [(4x \u2013 1) \/ (3)]<\/p>\n\n\n\n<p>Apply the specific point again.<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = [(4(1) \u2013 1) \/ (3)]<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = [(4 \u2013 1) \/ (3)]<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = [(3) \/ (3)]<\/p>\n\n\n\n<p>Lim<sub>x\u21921<\/sub> [(2x<sup>2<\/sup> \u2013 x \u2013 1) \/ (3x \u2013 3)] = 1<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"sum-up\"><\/span><strong>Sum Up<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Now you can grab all the basics of types and rules of limit calculus from this post as we have discussed almost everything about them along with formulas and examples.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The concept of limit calculus is one of the most essential concepts. Calculus is one of the most important concepts in mathematics, and the limit is one of its most fundamental concepts. Without it, calculus would be much more difficult to understand. In this article, we\u2019ll provide a brief overview of the limit and outline [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":15704,"comment_status":"open","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":"set","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":[137],"tags":[1993,1994,1992],"class_list":["post-15702","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","tag-rules-of-limit-calculus","tag-types-of-limit-calculus","tag-what-is-limit-calculus"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/15702","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/comments?post=15702"}],"version-history":[{"count":0,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/15702\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media\/15704"}],"wp:attachment":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media?parent=15702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/categories?post=15702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/tags?post=15702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}