{"id":38472,"date":"2026-05-26T07:56:09","date_gmt":"2026-05-26T07:56:09","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=38472"},"modified":"2026-05-26T08:41:24","modified_gmt":"2026-05-26T08:41:24","slug":"horizontal-asymptote","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/horizontal-asymptote\/","title":{"rendered":"Horizontal Asymptote: Rules, Formula, and Easy Examples"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"38472\" class=\"elementor elementor-38472\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<div class=\"elementor-inner\">\n\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-041a8ae elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"041a8ae\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ae1767a\" data-id=\"ae1767a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9d3173d elementor-widget elementor-widget-text-editor\" data-id=\"9d3173d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">For those working on algebra or graphing functions, chances are you have come across all kinds of material about horizontal asymptotes. A good understanding of a horizontal asymptote will provide an understanding of how to evaluate the behavior of a graph as approach very small or very large positive integer values of x.<\/span><\/p><p><strong>In this guide, you will cover the following:<\/strong><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><p><span style=\"font-weight: 400;\">What exactly is a horizontal asymptote?\u00a0<\/span><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><span style=\"font-weight: 400;\">Rules regarding horizontal asymptote(s).\u00a0<\/span><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><span style=\"font-weight: 400;\">How to determine the horizontal asymptote.\u00a0<\/span><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><span style=\"font-weight: 400;\">Examples of horizontal asymptotes broken down step-by-step.\u00a0<\/span><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><span style=\"font-weight: 400;\">Common errors associated with horizontal asymptotes.<\/span><\/p><\/li><\/ul><p><span style=\"font-weight: 400;\">Let\u2019s get started with the basics.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-3ca7d04 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3ca7d04\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a505cf9\" data-id=\"a505cf9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7b70028 elementor-widget elementor-widget-text-editor\" data-id=\"7b70028\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Horizontal Asymptotes<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cc210f7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cc210f7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3e8acd2\" data-id=\"3e8acd2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-85b4465 elementor-widget elementor-widget-text-editor\" data-id=\"85b4465\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">A horizontal asymptote is a horizontal line that a graph approaches when x goes to either positive infinity and negative infinity.<\/span><\/p><p><span style=\"font-weight: 400;\">In other words, the graph will eventually get near but will not reach a certain value and this is exactly why these asymptotes exist.<\/span><\/p><p><b>Common examples of horizontal asymptotes can be found in:<\/b><\/p><ul><li><span style=\"font-weight: 400;\">Rational Functions<\/span><\/li><li><span style=\"font-weight: 400;\">Exponential Functions<\/span><\/li><li><span style=\"font-weight: 400;\">Graphs from calculus<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">For example, in many of the graphs, there is a curved line that becomes level with a horizontal line.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-a614ea1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a614ea1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-18274b3\" data-id=\"18274b3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-86d170f elementor-widget elementor-widget-text-editor\" data-id=\"86d170f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Horizontal Asymptote Formula<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-72d41b6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"72d41b6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-69259c4\" data-id=\"69259c4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6fb9d24 elementor-widget elementor-widget-text-editor\" data-id=\"6fb9d24\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Rational Functions will typically have one or more horizontal asymptotes.<\/span><\/p><p><span style=\"font-weight: 400;\">Rational functions take the form shown below:<\/span><\/p><ul><li><span style=\"font-weight: 400;\">P(x) = numerator polynomial<\/span><\/li><li><span style=\"font-weight: 400;\">Q(x) = denominator polynomial<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">To determine whether a rational function has a horizontal asymptote, compare the degree of the numerator with the degree of the denominator.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-e5d90ed elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e5d90ed\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c55b0d2\" data-id=\"c55b0d2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5416825 elementor-widget elementor-widget-text-editor\" data-id=\"5416825\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Horizontal Asymptote Formula<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-b971541 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b971541\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9d2f0c9\" data-id=\"9d2f0c9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7b7cb1e elementor-widget elementor-widget-text-editor\" data-id=\"7b7cb1e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">There are three important horizontal asymptote rules.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-7913aa4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7913aa4\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-db7b7c7\" data-id=\"db7b7c7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3b2ee3a elementor-widget elementor-widget-text-editor\" data-id=\"3b2ee3a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4><b>Rule 1: Numerator Degree Is Smaller<\/b><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-46714d5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"46714d5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2138850\" data-id=\"2138850\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d338fe4 elementor-widget elementor-widget-text-editor\" data-id=\"d338fe4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is<strong>:<br \/><\/strong><\/span><strong><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/semantics><\/math><\/span><\/strong><\/p><p><b>Example<\/b><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-611eeba elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"611eeba\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-06cd502\" data-id=\"06cd502\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-566b3d8 elementor-widget elementor-widget-html\" data-id=\"566b3d8\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"math-card\">\n\n  <h2>Rule 1: Numerator Degree Smaller<\/h2>\n\n  <input type=\"range\" min=\"1\" max=\"10\" value=\"3\" class=\"slider\" id=\"sliderRule1\"\/>\n\n  <div class=\"graph-box\">\n\n    <div class=\"horizontal-line\"><\/div>\n\n    <div class=\"curve curve1\" id=\"curveRule1\"><\/div>\n\n    <div class=\"label asymptote-label\">\n      y = 0\n    <\/div>\n\n  <\/div>\n\n  <table>\n    <tr>\n      <th>Function<\/th>\n      <td>f(x) = 3 \/ (x\u00b2 + 1)<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Numerator Degree<\/th>\n      <td>0<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Denominator Degree<\/th>\n      <td>2<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Horizontal Asymptote<\/th>\n      <td>y = 0<\/td>\n    <\/tr>\n  <\/table>\n\n<\/div>\n\n<style>\n.math-card{\n  max-width:550px;\n  margin:auto;\n  background:#fff;\n  padding:20px;\n  border-radius:18px;\n  box-shadow:0 10px 25px rgba(0,0,0,0.08);\n  font-family:Arial,sans-serif;\n}\n\n.slider{\n  width:100%;\n  margin-bottom:20px;\n}\n\n.graph-box{\n  width:100%;\n  height:250px;\n  background:#f8fbff;\n  border-radius:14px;\n  position:relative;\n  overflow:hidden;\n  margin-bottom:20px;\n}\n\n.horizontal-line{\n  position:absolute;\n  width:100%;\n  height:3px;\n  background:#ef4444;\n  bottom:30px;\n}\n\n.curve1{\n  position:absolute;\n  width:220px;\n  height:220px;\n  border:5px solid #2563eb;\n  border-color:#2563eb transparent transparent transparent;\n  border-radius:50%;\n  left:40px;\n  top:10px;\n}\n\n.label{\n  position:absolute;\n  bottom:8px;\n  right:20px;\n  color:#ef4444;\n  font-weight:bold;\n}\n\ntable{\n  width:100%;\n  border-collapse:collapse;\n}\n\ntable th,\ntable td{\n  border:1px solid #ddd;\n  padding:12px;\n}\n\ntable th{\n  background:#f3f4f6;\n}\n<\/style>\n\n<script>\nconst sliderRule1 = document.getElementById(\"sliderRule1\");\n\nsliderRule1.oninput = function(){\n\n  let size = 150 + (this.value * 10);\n\n  document.getElementById(\"curveRule1\").style.width = size + \"px\";\n  document.getElementById(\"curveRule1\").style.height = size + \"px\";\n};\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-037dda5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"037dda5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7d594d5\" data-id=\"7d594d5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b9e43f1 elementor-widget elementor-widget-text-editor\" data-id=\"b9e43f1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Numerator degree = 0<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Denominator degree = 2<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Since the numerator degree is smaller:<\/span><\/p><p><span style=\"font-weight: 400;\">Horizontal asymptote = y = 0<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-9e73448 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9e73448\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4e68048\" data-id=\"4e68048\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-92cdf38 elementor-widget elementor-widget-text-editor\" data-id=\"92cdf38\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Rule 2: Degrees Are Equal<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-1d84688 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1d84688\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cac3181\" data-id=\"cac3181\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d1d0d36 elementor-widget elementor-widget-text-editor\" data-id=\"d1d0d36\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p data-start=\"1809\" data-end=\"1896\">If the numerator and denominator have the same degree, divide the leading coefficients.<\/p><p data-section-id=\"48h0qw\" data-start=\"1898\" data-end=\"1909\"><strong>Formula<\/strong><\/p><ul><li data-start=\"1911\" data-end=\"1948\"><span class=\"inline-block align-middle\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">y=\\frac{a}{b}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li><\/ul><p data-start=\"1950\" data-end=\"1956\">Where:<\/p><ul><li>a = leading coefficient of numerator<\/li><li>b = leading coefficient of denominator<\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-fb02972 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fb02972\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-dcb1efe\" data-id=\"dcb1efe\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a65b9bb elementor-widget elementor-widget-html\" data-id=\"a65b9bb\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"math-card\">\n\n  <h2>Rule 2: Equal Degrees<\/h2>\n\n  <input type=\"range\" min=\"1\" max=\"5\" value=\"2\" class=\"slider\" id=\"sliderRule2\"\/>\n\n  <div class=\"graph-box\">\n\n    <div class=\"horizontal-line middle-line\"><\/div>\n\n    <div class=\"curve2\"><\/div>\n\n    <div class=\"label asymptote-label2\">\n      y = <span id=\"asymValue2\">2<\/span>\n    <\/div>\n\n  <\/div>\n\n  <table>\n    <tr>\n      <th>Function<\/th>\n      <td>f(x) = (4x\u00b2 + 1)\/(2x\u00b2 \u2212 3)<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Leading Coefficients<\/th>\n      <td>4 \u00f7 2<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Horizontal Asymptote<\/th>\n      <td>y = <span id=\"tableAsym2\">2<\/span><\/td>\n    <\/tr>\n  <\/table>\n\n<\/div>\n\n<style>\n.math-card{\n  max-width:550px;\n  margin:auto;\n  background:#fff;\n  padding:20px;\n  border-radius:18px;\n  box-shadow:0 10px 25px rgba(0,0,0,0.08);\n  font-family:Arial,sans-serif;\n}\n\n.slider{\n  width:100%;\n  margin-bottom:20px;\n}\n\n.graph-box{\n  width:100%;\n  height:250px;\n  background:#f8fbff;\n  border-radius:14px;\n  position:relative;\n  overflow:hidden;\n  margin-bottom:20px;\n}\n\n.middle-line{\n  position:absolute;\n  width:100%;\n  height:3px;\n  background:#ef4444;\n  top:50%;\n}\n\n.curve2{\n  position:absolute;\n  width:250px;\n  height:180px;\n  border:5px solid #7c3aed;\n  border-color:#7c3aed transparent transparent transparent;\n  border-radius:50%;\n  left:20px;\n  top:40px;\n}\n\n.asymptote-label2{\n  position:absolute;\n  top:20px;\n  right:20px;\n  color:#ef4444;\n  font-weight:bold;\n}\n\ntable{\n  width:100%;\n  border-collapse:collapse;\n}\n\ntable th,\ntable td{\n  border:1px solid #ddd;\n  padding:12px;\n}\n\ntable th{\n  background:#f3f4f6;\n}\n<\/style>\n\n<script>\nconst sliderRule2 = document.getElementById(\"sliderRule2\");\n\nsliderRule2.oninput = function(){\n\n  let value = this.value;\n\n  document.getElementById(\"asymValue2\").innerHTML = value;\n  document.getElementById(\"tableAsym2\").innerHTML = value;\n};\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-8119d72 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8119d72\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c02a11f\" data-id=\"c02a11f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-718dcc1 elementor-widget elementor-widget-text-editor\" data-id=\"718dcc1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p data-start=\"2093\" data-end=\"2114\">Leading coefficients:<\/p><ul data-start=\"2115\" data-end=\"2148\"><li data-section-id=\"k1xaw8\" data-start=\"2115\" data-end=\"2130\">Numerator = 4<\/li><li data-section-id=\"14obvnp\" data-start=\"2131\" data-end=\"2148\">Denominator = 2<\/li><\/ul><p data-start=\"2150\" data-end=\"2153\">So<strong>: <\/strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y=2<\/annotation><\/semantics><\/math><\/p><p data-start=\"2194\" data-end=\"2228\">The horizontal asymptote is y = 2.<\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-453a245 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"453a245\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-42b65e6\" data-id=\"42b65e6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-37ba6c5 elementor-widget elementor-widget-text-editor\" data-id=\"37ba6c5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4><b>Rule 3: Numerator Degree Is Larger<\/b><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-9bfe094 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9bfe094\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c37aa8b\" data-id=\"c37aa8b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9875b04 elementor-widget elementor-widget-text-editor\" data-id=\"9875b04\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p data-start=\"2274\" data-end=\"2371\">If the numerator degree is greater than the denominator degree, there is no horizontal asymptote.<\/p><p data-section-id=\"16zgw4u\" data-start=\"2373\" data-end=\"2384\"><strong>Example<\/strong><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-812a8f9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"812a8f9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bc4b553\" data-id=\"bc4b553\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2f0d644 elementor-widget elementor-widget-html\" data-id=\"2f0d644\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"math-card\">\n\n  <h2>Rule 3: Numerator Degree Larger<\/h2>\n\n  <input type=\"range\" min=\"2\" max=\"8\" value=\"4\" class=\"slider\" id=\"sliderRule3\"\/>\n\n  <div class=\"graph-box\">\n\n    <div class=\"curve3\" id=\"curveRule3\"><\/div>\n\n    <div class=\"no-asymptote\">\n      No Horizontal Asymptote\n    <\/div>\n\n  <\/div>\n\n  <table>\n    <tr>\n      <th>Function<\/th>\n      <td>f(x) = (x\u00b3 + 1)\/(x\u00b2 + 2)<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Numerator Degree<\/th>\n      <td>3<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Denominator Degree<\/th>\n      <td>2<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Result<\/th>\n      <td>No Horizontal Asymptote<\/td>\n    <\/tr>\n  <\/table>\n\n<\/div>\n\n<style>\n.math-card{\n  max-width:550px;\n  margin:auto;\n  background:#fff;\n  padding:20px;\n  border-radius:18px;\n  box-shadow:0 10px 25px rgba(0,0,0,0.08);\n  font-family:Arial,sans-serif;\n}\n\n.slider{\n  width:100%;\n  margin-bottom:20px;\n}\n\n.graph-box{\n  width:100%;\n  height:250px;\n  background:#f8fbff;\n  border-radius:14px;\n  position:relative;\n  overflow:hidden;\n  margin-bottom:20px;\n}\n\n.curve3{\n  position:absolute;\n  width:220px;\n  height:220px;\n  border-left:5px solid #f59e0b;\n  border-bottom:5px solid #f59e0b;\n  border-radius:0 0 0 100%;\n  left:40px;\n  top:20px;\n}\n\n.no-asymptote{\n  position:absolute;\n  bottom:15px;\n  left:15px;\n  color:#f59e0b;\n  font-weight:bold;\n}\n\ntable{\n  width:100%;\n  border-collapse:collapse;\n}\n\ntable th,\ntable td{\n  border:1px solid #ddd;\n  padding:12px;\n}\n\ntable th{\n  background:#f3f4f6;\n}\n<\/style>\n\n<script>\nconst sliderRule3 = document.getElementById(\"sliderRule3\");\n\nsliderRule3.oninput = function(){\n\n  let size = 120 + (this.value * 15);\n\n  document.getElementById(\"curveRule3\").style.width = size + \"px\";\n  document.getElementById(\"curveRule3\").style.height = size + \"px\";\n};\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ec80c9c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ec80c9c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-31d9a41\" data-id=\"31d9a41\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8950848 elementor-widget elementor-widget-text-editor\" data-id=\"8950848\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>How to Find the Horizontal Asymptote<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-2ca9beb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2ca9beb\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-56cbc03\" data-id=\"56cbc03\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-70762db elementor-widget elementor-widget-text-editor\" data-id=\"70762db\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4><b>Step 1: Identify the Rational Function<\/b><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-8fd1a1c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8fd1a1c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b079549\" data-id=\"b079549\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6f42407 elementor-widget elementor-widget-image\" data-id=\"6f42407\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-image\">\n\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1024\" height=\"319\" src=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2026\/05\/horizontal-asymptote-2.webp\" class=\"attachment-large size-large wp-image-38499\" alt=\"Horizontal Asymptote\" loading=\"lazy\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-6ca93ab elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6ca93ab\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9ca4db0\" data-id=\"9ca4db0\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4626048 elementor-widget elementor-widget-text-editor\" data-id=\"4626048\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4><b>Step 2: Find the Degree<\/b><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-1bed26f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1bed26f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9476ef5\" data-id=\"9476ef5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9aa9780 elementor-widget elementor-widget-text-editor\" data-id=\"9aa9780\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">The degree is the highest exponent.<\/span><\/p><p><span style=\"font-weight: 400;\">Examples:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x\u00b2 has degree 2<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x\u00b3 has degree 3<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">For the function above:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Numerator degree = 1<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Denominator degree = 1<\/span><\/li><\/ul><p>Recommended Reading: <a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/how-to-find-the-radius-of-a-circle\/\">How to Find the Radius of a Circle: Easy Formulas and Examples<\/a><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f895d12 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f895d12\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ff5e49a\" data-id=\"ff5e49a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bdb68c2 elementor-widget elementor-widget-text-editor\" data-id=\"bdb68c2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4><b>Step 3: Apply the Correct Rule<\/b><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-68cbbc4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68cbbc4\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-137360e\" data-id=\"137360e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-29c6178 elementor-widget elementor-widget-text-editor\" data-id=\"29c6178\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Since the degrees are equal, divide the leading coefficients.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Leading coefficient of numerator = 2<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Leading coefficient of denominator = 1<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So:<\/span><\/p><p><span style=\"font-weight: 400;\">y=2<\/span><\/p><p><span style=\"font-weight: 400;\">The horizontal asymptote is y = 2.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-6fa343e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6fa343e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fbe4153\" data-id=\"fbe4153\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4c82636 elementor-widget elementor-widget-text-editor\" data-id=\"4c82636\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Horizontal Asymptote vs Vertical Asymptote<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-8bf3e23 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8bf3e23\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-60f89a3\" data-id=\"60f89a3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7676e38 elementor-widget elementor-widget-html\" data-id=\"7676e38\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"math-card\">\n\n  <h2>Horizontal vs Vertical Asymptote<\/h2>\n\n  <input type=\"range\" min=\"1\" max=\"10\" value=\"5\" class=\"slider\" id=\"sliderRule4\"\/>\n\n  <div class=\"graph-box\">\n\n    <div class=\"horizontal-line middle-line\"><\/div>\n\n    <div class=\"vertical-line\" id=\"verticalLine\"><\/div>\n\n    <div class=\"curve4\"><\/div>\n\n    <div class=\"h-label\">\n      Horizontal\n    <\/div>\n\n    <div class=\"v-label\">\n      Vertical\n    <\/div>\n\n  <\/div>\n\n  <table>\n    <tr>\n      <th>Horizontal Asymptote<\/th>\n      <td>y = number<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Vertical Asymptote<\/th>\n      <td>x = number<\/td>\n    <\/tr>\n\n    <tr>\n      <th>Purpose<\/th>\n      <td>Shows graph behavior<\/td>\n    <\/tr>\n  <\/table>\n\n<\/div>\n\n<style>\n.math-card{\n  max-width:550px;\n  margin:auto;\n  background:#fff;\n  padding:20px;\n  border-radius:18px;\n  box-shadow:0 10px 25px rgba(0,0,0,0.08);\n  font-family:Arial,sans-serif;\n}\n\n.slider{\n  width:100%;\n  margin-bottom:20px;\n}\n\n.graph-box{\n  width:100%;\n  height:250px;\n  background:#f8fbff;\n  border-radius:14px;\n  position:relative;\n  overflow:hidden;\n  margin-bottom:20px;\n}\n\n.middle-line{\n  position:absolute;\n  width:100%;\n  height:3px;\n  background:#ef4444;\n  top:50%;\n}\n\n.vertical-line{\n  position:absolute;\n  width:3px;\n  height:100%;\n  background:#10b981;\n  left:50%;\n}\n\n.curve4{\n  position:absolute;\n  width:220px;\n  height:220px;\n  border:5px solid #2563eb;\n  border-color:#2563eb transparent transparent transparent;\n  border-radius:50%;\n  left:40px;\n  top:20px;\n}\n\n.h-label{\n  position:absolute;\n  top:55%;\n  right:15px;\n  color:#ef4444;\n  font-weight:bold;\n}\n\n.v-label{\n  position:absolute;\n  top:10px;\n  left:53%;\n  color:#10b981;\n  font-weight:bold;\n}\n\ntable{\n  width:100%;\n  border-collapse:collapse;\n}\n\ntable th,\ntable td{\n  border:1px solid #ddd;\n  padding:12px;\n}\n\ntable th{\n  background:#f3f4f6;\n}\n<\/style>\n\n<script>\nconst sliderRule4 = document.getElementById(\"sliderRule4\");\n\nsliderRule4.oninput = function(){\n\n  let position = 20 + (this.value * 15);\n\n  document.getElementById(\"verticalLine\").style.left = position + \"px\";\n};\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-3a61763 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3a61763\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-dde815f\" data-id=\"dde815f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2b0474e elementor-widget elementor-widget-text-editor\" data-id=\"2b0474e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<table><tbody><tr><td><p><b>Horizontal Asymptote<\/b><\/p><\/td><td><p><b>Vertical Asymptote<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Written as y = number<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Written as x = number<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Shows end behavior<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Shows undefined points<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Found using degrees<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Found using denominator zeros<\/span><\/p><br \/><br \/><\/td><\/tr><\/tbody><\/table>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-3ba21d8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3ba21d8\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0f8d997\" data-id=\"0f8d997\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ff0a380 elementor-widget elementor-widget-text-editor\" data-id=\"ff0a380\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Can a Graph Cross a Horizontal Asymptote?<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-8e3d3bc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8e3d3bc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d64599c\" data-id=\"d64599c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fefd3da elementor-widget elementor-widget-text-editor\" data-id=\"fefd3da\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Yes.\u00a0<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Sometimes a graph will cross a horizontal asymptote.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">A horizontal asymptote only describes the behavior of the graph when the x-values are very large or very small.<\/span><\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-bffb613 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bffb613\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d291703\" data-id=\"d291703\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b5f806e elementor-widget elementor-widget-text-editor\" data-id=\"b5f806e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 data-section-id=\"8s1q7u\" data-start=\"4386\" data-end=\"4412\"><strong>Common Mistakes to Avoid<\/strong><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-23b3f14 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"23b3f14\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e3f9438\" data-id=\"e3f9438\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f0ba662 elementor-widget elementor-widget-text-editor\" data-id=\"f0ba662\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><b>1. Always compare the highest powers first.<\/b><\/p><p><span style=\"font-weight: 400;\">Using wrong coefficients<\/span><\/p><p><b>2. When degrees are the same, use only the leading coefficients.<\/b><\/p><p><span style=\"font-weight: 400;\">Vertical and Horizontal Asymptotes Confused<\/span><\/p><p><b>3. Keep in mind:<\/b><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Horizontal Asymptotes Use Y Vertical Asymptotes Use X Think Graph Never Touches Asymptote<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Some graphs may cross horizontal asymptotes.<\/span><\/li><\/ul><p><b>4. Real World Applications of Horizontal Asymptotes<br \/><br \/><\/b><span style=\"font-weight: 400;\">Horizontal asymptotes are used for:<\/span><b><\/b><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Models of Population Growth<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Physics formulas<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Graphs of economics<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Mathematics for Engineering<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Higher Mathematics and Calculus<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">They help predict behaviour long term in graphs and equations.<\/span><b><\/b><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-7b0f57f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7b0f57f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8a5ae73\" data-id=\"8a5ae73\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e705edd elementor-widget elementor-widget-text-editor\" data-id=\"e705edd\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Conclusion<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c801e0c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c801e0c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-50392f5\" data-id=\"50392f5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ffedfd4 elementor-widget elementor-widget-text-editor\" data-id=\"ffedfd4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Once you know the basic rules, it&#8217;s easy to understand the horizontal asymptote. By comparing the degrees of the numerator and denominator, you can readily determine whether a function has a horizontal asymptote and what its equation is.<\/span><\/p><p><span style=\"font-weight: 400;\">Practice recognizing and graphing various rational functions.<br \/><\/span><\/p><p><span style=\"font-weight: 400;\">Want to excite your child about math and sharpen their math skills? Moonpreneur&#8217;s online math curriculum is unique as it helps children understand math skills through hands-on lessons, assists them in building real-life applications, and excites them to learn math.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">You can opt for our <a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\">Advanced Math<\/a> or Vedic Math+Mental Math courses. Our <a href=\"https:\/\/mp.moonpreneur.com\/math-quiz-for-kids\/\">Math Quiz<\/a> for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.<br \/><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-b160a49 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b160a49\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7585324\" data-id=\"7585324\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-78292a5 elementor-widget elementor-widget-toggle\" data-id=\"78292a5\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1251\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1251\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">What is an asymptote?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1251\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1251\"><p><span style=\"font-weight: 400;\">A graph has a horizontal asymptote when the graph approaches a horizontal line as x becomes very large or very small.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1252\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1252\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">2. How do you obtain the horizontal asymptote?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1252\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1252\"><p><span style=\"font-weight: 400;\">Compare the degree of the numerator and the degree of the denominator and use the proper rule.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1253\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1253\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">What are the rules for horizontal asymptotes?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1253\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1253\"><p><span style=\"font-weight: 400;\">Smaller numerator degree \u2192 y = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Equal degrees \u2192 divide leading coefficients<\/span><\/p><p><span style=\"font-weight: 400;\">Larger numerator degree \u2192 no horizontal asymptote<\/span><\/p><p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1254\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-1254\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\"> 4. Can a graph cross horizontal asymptotes ?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1254\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-1254\"><p><span style=\"font-weight: 400;\">Yes, some graphs cross horizontal asymptotes.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1255\" class=\"elementor-tab-title\" data-tab=\"5\" role=\"button\" aria-controls=\"elementor-tab-content-1255\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">What's the difference between vertical and horizontal asymptotes?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1255\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"5\" role=\"region\" aria-labelledby=\"elementor-tab-title-1255\"><p><span style=\"font-weight: 400;\">Horizontal asymptotes describe what happens at the ends of a graph, while vertical asymptotes indicate where a function is undefined.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>For those working on algebra or graphing functions, chances are you have come across all kinds of material about horizontal asymptotes. A good understanding of a horizontal asymptote will provide an understanding of how to evaluate the behavior of a graph as approach very small or very large positive integer values of x. In this [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":38492,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[979],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38472"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/comments?post=38472"}],"version-history":[{"count":22,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38472\/revisions"}],"predecessor-version":[{"id":38503,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38472\/revisions\/38503"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/38492"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=38472"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=38472"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=38472"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}