{"id":29667,"date":"2026-10-27T07:02:48","date_gmt":"2026-10-27T07:02:48","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=29667"},"modified":"2026-10-27T08:46:05","modified_gmt":"2026-10-27T08:46:05","slug":"what-is-the-value-of-e-power-minus-infinity","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/math-explorer-program\/what-is-the-value-of-e-power-minus-infinity\/","title":{"rendered":"What is the value of e power minus infinity ?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"29667\" class=\"elementor elementor-29667\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-085098e elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"085098e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a433c8c\" data-id=\"a433c8c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-349994b\" data-id=\"349994b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f34100d elementor-widget elementor-widget-heading\" data-id=\"f34100d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">What Is the Value of e Power Minus Infinity?\n<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1e05c0e elementor-widget elementor-widget-text-editor\" data-id=\"1e05c0e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Chances are you ran into e^(\u2212\u221e) in a calculus problem and weren&#8217;t sure what to do with it. Here&#8217;s the quick answer: the value of e power minus infinity is <\/span><b>0<\/b><span style=\"font-weight: 400;\">. The longer answer is more interesting. Infinity isn&#8217;t a number you can plug in, so e^(\u2212\u221e) is really shorthand for a question: what happens to be raised to a power when that power keeps dropping further below zero? The result shrinks toward 0 and gets as close as you like, but it never dips below it. Let&#8217;s walk through why, one step at a time.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bb9aba4 elementor-align-center elementor-widget elementor-widget-button\" data-id=\"bb9aba4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2b6f104 elementor-widget elementor-widget-image\" data-id=\"2b6f104\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"647\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math-1024x647.webp\" class=\"attachment-large size-large wp-image-29777\" alt=\"Value of e power minus infinity explained with a student studying exponential decay in mathematics.\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math-1024x647.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math-300x190.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math-768x486.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math-1536x971.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Thoughtful-Student-and-Chalkboard-Math.webp 1577w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-10c4991\" data-id=\"10c4991\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c3203db elementor-widget elementor-widget-html\" data-id=\"c3203db\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; \r\n    z-index: 1 !important;\r\n    right: 70px;\r\n    width: 100%;\r\n    max-width: 340px;\r\n    font-family: 'Poppins', sans-serif;\r\n}\r\n\r\n\/* Image Styling *\/\r\n.toc-sticky-image {\r\n    width: 100%;\r\n    height: auto;\r\n    border-radius: 22px; \r\n    margin-bottom: 15px; \r\n    box-shadow: 0 4px 12px rgba(0,0,0,0.05);\r\n    display: block;\r\n}\r\n\r\n\/* TOC Container *\/\r\n.custom-toc {\r\n    background: #fff;\r\n    border: 1px solid #dfe3ea;\r\n    border-radius: 22px;\r\n    padding: 0px 28px;\r\n    width: 100%;\r\n    max-height: 24rem; \r\n    overflow-y: auto;\r\n    box-shadow: 0 2px 8px rgba(0,0,0,0.03);\r\n}\r\n\r\n\/* Title *\/\r\n.custom-toc .toc-title {\r\n    font-size: 25px;\r\n    font-weight: 700;\r\n    color: #17233c;\r\n    position: sticky;\r\n    padding: 0.6rem 0rem;\r\n    background-color: white;\r\n    top: 0px;\r\n    margin-bottom: 12px;\r\n    display: flex;\r\n    align-items: center;\r\n    gap: 10px;\r\n    z-index: 2;\r\n}\r\n\r\n\/* List *\/\r\n.custom-toc .toc-list {\r\n    list-style: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\n\/* Links *\/\r\n.custom-toc .toc-list li {\r\n    margin-bottom: 22px;\r\n}\r\n\r\n.custom-toc .toc-list a {\r\n    text-decoration: none;\r\n    color: #6b7a99;\r\n    font-size: 14px;\r\n    font-weight: 600;\r\n    transition: 0.3s ease;\r\n    display: inline-block;\r\n}\r\n\r\n\/* Hover Color *\/\r\n.custom-toc .toc-list a:hover {\r\n    color: #8B0000;\r\n}\r\n\r\n\/* H2 Indent *\/\r\n.custom-toc .toc-list .toc-h2 {\r\n    padding-left: 18px;\r\n    font-size: 14px;\r\n}\r\n\r\n\/* Mobile Responsive Adjustments *\/\r\n@media(max-width:768px) {\r\n    .toc-wrapper {\r\n        position: relative;\r\n        top: 0;\r\n        z-index: 1;\r\n        right: 0;\r\n        max-width: 100%;\r\n        margin-bottom: 30px;\r\n    }\r\n\r\n    .custom-toc {\r\n        max-height: none; \r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-353e20b elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"353e20b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4a309fe\" data-id=\"4a309fe\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7121f4d\" data-id=\"7121f4d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1beebe5 elementor-widget elementor-widget-heading\" data-id=\"1beebe5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">A Quick Refresher on the Number e\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fb08552 elementor-widget elementor-widget-text-editor\" data-id=\"fb08552\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The letter e stands for Euler&#8217;s number, roughly 2.71828. It&#8217;s irrational, so its decimals run on forever without a pattern, a bit like pi. You&#8217;ll see it in compound interest, population growth, radioactive decay and probability, which is why it gets so much attention in exams.<\/span><\/p><p><span style=\"font-weight: 400;\">Only two facts about the function f(x) = e\u02e3 matter for this topic:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">It is always positive. Whatever real number you put in, e\u02e3 comes out above zero.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">It keeps climbing. Bigger x means bigger e\u02e3.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Hold on to those two points. The whole answer grows out of them.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-af4b735\" data-id=\"af4b735\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-59697f1 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"59697f1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c15947d\" data-id=\"c15947d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-63cd2dd\" data-id=\"63cd2dd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bbc32c9 elementor-widget elementor-widget-heading\" data-id=\"bbc32c9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Infinity Is a Direction, Not a Number\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-408fb71 elementor-widget elementor-widget-text-editor\" data-id=\"408fb71\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This is where many students get tripped up. You can&#8217;t treat \u221e like 5 or 100. It&#8217;s a way of saying &#8220;keeps growing with no upper limit.&#8221; So when a textbook writes e^(\u2212\u221e), it&#8217;s compressing a longer sentence: &#8220;let the exponent become more and more negative and see where the value is heading.&#8221;<\/span><\/p><p><span style=\"font-weight: 400;\">That&#8217;s exactly why serious maths writes this idea as a limit. Think of it as describing a journey instead of a destination you can stand on.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-38a891a\" data-id=\"38a891a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f1c3864 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f1c3864\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-358d789\" data-id=\"358d789\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-25d55bb\" data-id=\"25d55bb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2b767d5 elementor-widget elementor-widget-heading\" data-id=\"2b767d5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What Is e to the Power of Negative Infinity?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e17e7a4 elementor-widget elementor-widget-text-editor\" data-id=\"e17e7a4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">It equals 0. To see why, start small and watch the pattern:<\/span><\/p><table><tbody><tr><td><p><b>Exponent<\/b><\/p><\/td><td><p><b>Value of e raised to it (approx.)<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u22121<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.3679<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u22125<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.006738<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u221210<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.0000454<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u221220<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">0.00000000206<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u221250<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1.93 \u00d7 10\u207b\u00b2\u00b2<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">\u2212100<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3.72 \u00d7 10\u207b\u2074\u2074<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">Each time the exponent drops, the answer collapses. By \u2212100 the value is so tiny that most calculators just show it in scientific notation. Keep going and the numbers keep sliding toward zero, which is what we mean when we say e to the power of negative infinity is 0.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-cec5840\" data-id=\"cec5840\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b828bfb elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"b828bfb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8e5ad6e\" data-id=\"8e5ad6e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-dfa85cc\" data-id=\"dfa85cc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-da74aab elementor-widget elementor-widget-heading\" data-id=\"da74aab\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What Is the Value of e^{-\u221e}?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-61579b7 elementor-widget elementor-widget-text-editor\" data-id=\"61579b7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">There&#8217;s a second way to get the same answer, and many students find it easier to believe. A negative exponent means &#8220;flip it&#8221;:<\/span><\/p><p><span style=\"font-weight: 400;\">e^{-\\infty} = \\frac{1}{e^{\\infty}}<\/span><\/p><p><span style=\"font-weight: 400;\">Now ask what e^\u221e is. It&#8217;s enormous, with no ceiling at all. So you&#8217;re dividing 1 by something that grows without limit. Cut a cake into more and more slices and each slice gets smaller and smaller; that&#8217;s the same idea. A share that keeps shrinking heads for zero, and the value of e^{-\u221e} is 0.<\/span><\/p><p><span style=\"font-weight: 400;\">Two roads, one answer. That&#8217;s usually a good sign in maths.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-eae1564\" data-id=\"eae1564\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8ae0545 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"8ae0545\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4ad7aa3\" data-id=\"4ad7aa3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ab22ab9\" data-id=\"ab22ab9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6692dd3 elementor-widget elementor-widget-heading\" data-id=\"6692dd3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Evaluate the Limit of e^{-x} as x Approaches Infinity\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fe63784 elementor-widget elementor-widget-text-editor\" data-id=\"fe63784\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Exam questions rarely say &#8220;e to the minus infinity.&#8221; They ask you to evaluate the limit of e^{-x} as x approaches infinity, and here&#8217;s how to write that up cleanly.<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Rewrite the expression using the negative exponent rule: e\u207b\u02e3 = 1\/e\u02e3.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Look at the bottom of the fraction. As x \u2192 \u221e, e\u02e3 \u2192 \u221e.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The top stays fixed at 1, so you have 1 divided by a quantity that grows without bound.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">That gives 0.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">\\lim_{x \\to \\infty} e^{-x} = \\lim_{x \\to \\infty} \\frac{1}{e^{x}} = 0<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-f8fde63\" data-id=\"f8fde63\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bcd2e0e elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"bcd2e0e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b19bc5d\" data-id=\"b19bc5d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-27446b1\" data-id=\"27446b1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-474d328 elementor-widget elementor-widget-heading\" data-id=\"474d328\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The formal proof, if your syllabus asks for it\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a6a22eb elementor-widget elementor-widget-text-editor\" data-id=\"a6a22eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Higher-level courses want the epsilon argument. It sounds scary, but it&#8217;s short. Pick any tiny positive number \u03b5 less than 1. We want e\u207b\u02e3 &lt; \u03b5.<\/span><\/p><p><span style=\"font-weight: 400;\">Take the natural log of both sides. Since ln is increasing, and \u2212x flips the inequality when we multiply by \u22121, we get x &gt; ln(1\/\u03b5). So once x is bigger than N = ln(1\/\u03b5), the value of e\u207b\u02e3 is below \u03b5, and that holds no matter how small \u03b5 is. That is exactly the definition of the limit being 0.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1cbe73f\" data-id=\"1cbe73f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8e5a86c elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"8e5a86c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c7f66dd\" data-id=\"c7f66dd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c2388a2\" data-id=\"c2388a2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0f6a297 elementor-widget elementor-widget-heading\" data-id=\"0f6a297\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What Does e Raised to Minus Infinity Equal?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-33c03a5 elementor-widget elementor-widget-text-editor\" data-id=\"33c03a5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In one word, zero. But it&#8217;s a particular kind of zero, and the difference shows up in exams.<\/span><\/p><p><span style=\"font-weight: 400;\">For every real number x, e\u207b\u02e3 is strictly bigger than 0. The function never lands on zero; it just keeps closing the gap. Picture walking toward a wall by covering half the remaining distance each step. You get closer forever, yet you never touch the wall. In graph language, the line y = 0 is a horizontal asymptote.<\/span><\/p><p><span style=\"font-weight: 400;\">The graph of y = e\u02e3 makes this easy to picture. On the right it shoots upward. On the left it flattens out and hugs the x-axis, and it passes through (0, 1) because e\u2070 = 1. The graph of y = e\u207b\u02e3 is its mirror image, sliding down toward the x-axis as you move right.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-679dc82\" data-id=\"679dc82\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-754e6d7 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"754e6d7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-57220fa\" data-id=\"57220fa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-16a5bea\" data-id=\"16a5bea\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-90acf32 elementor-widget elementor-widget-heading\" data-id=\"90acf32\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What is lim x\u2192\u221e e^{-x}?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e556e7c elementor-widget elementor-widget-text-editor\" data-id=\"e556e7c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">That expression, lim x\u2192\u221e e^{-x}, is just the standard shorthand for the limit we solved above, read aloud as &#8220;the limit of e to the minus x, as x tends to infinity.&#8221; The answer is 0.<\/span><\/p><p><span style=\"font-weight: 400;\">It&#8217;s worth knowing how this connects to two neighbours, because they get mixed up all the time:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">lim (x\u2192\u2212\u221e) e\u02e3 = 0. Same behaviour, written with the sign moved into x itself.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">lim (x\u2192\u221e) e\u02e3 = \u221e. Flip the sign of the exponent and the answer flips from tiny to huge.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So the sign of the exponent decides everything. Large and negative sends the value to 0; large and positive sends it to infinity.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-78656d8\" data-id=\"78656d8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c92e835 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"c92e835\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9db807b\" data-id=\"9db807b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6205caf\" data-id=\"6205caf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9df6777 elementor-widget elementor-widget-heading\" data-id=\"9df6777\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Worked Examples\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c0823c5 elementor-widget elementor-widget-text-editor\" data-id=\"c0823c5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><b>Example 1.<\/b><span style=\"font-weight: 400;\"> Find lim (x\u2192\u221e) 5e\u207b\u00b2\u02e3. As x grows, \u22122x heads to \u2212\u221e, so e\u207b\u00b2\u02e3 \u2192 0. Multiply by 5 and you still get <\/span><b>0<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Example 2.<\/b><span style=\"font-weight: 400;\"> Find lim (x\u2192\u221e) (3 + e\u207b\u02e3). The e\u207b\u02e3 part vanishes, leaving <\/span><b>3<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Example 3.<\/b><span style=\"font-weight: 400;\"> Find lim (x\u2192\u221e) x\u00b7e\u207b\u02e3. This looks like \u221e \u00d7 0, which isn&#8217;t an answer by itself. Rewrite it as x\/e\u02e3, which is \u221e\/\u221e, and use L&#8217;H\u00f4pital&#8217;s rule: the derivatives give 1\/e\u02e3, which tends to <\/span><b>0<\/b><span style=\"font-weight: 400;\">. In plain words, exponential decay wins the tug of war against x.<\/span><\/p><p><b>Example 4.<\/b><span style=\"font-weight: 400;\"> Find lim (x\u2192\u221e) 1\/(1 + e\u207b\u02e3). With e\u207b\u02e3 \u2192 0 you get 1\/(1 + 0) = <\/span><b>1<\/b><span style=\"font-weight: 400;\">. This is the well-known sigmoid curve, which levels off at 1.<\/span><\/p><p><span style=\"font-weight: 400;\">Try these yourself: lim (x\u2192\u221e) 7e\u207b\u02e3, lim (x\u2192\u221e) (4 \u2212 e\u207b\u02e3), and lim (x\u2192\u221e) e^(\u2212x\u00b2). The answers are 0, 4 and 0.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e795850\" data-id=\"e795850\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1b8ae44 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"1b8ae44\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a7952df\" data-id=\"a7952df\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7841745\" data-id=\"7841745\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d202c33 elementor-widget elementor-widget-heading\" data-id=\"d202c33\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Mistakes That Cost Marks\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8afeacd elementor-widget elementor-widget-text-editor\" data-id=\"8afeacd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Treating \u221e as an ordinary number.<\/b><span style=\"font-weight: 400;\"> Don&#8217;t substitute it like you would 5. Think in limits.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Expecting a negative answer.<\/b><span style=\"font-weight: 400;\"> The exponent is negative, but e\u02e3 is always positive, so the result approaches 0 from above.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Mixing up e\u207b\u02e3 and \u2212e\u02e3.<\/b><span style=\"font-weight: 400;\"> The first tends to 0 as x \u2192 \u221e. The second heads off to \u2212\u221e.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Saying it eventually equals 0.<\/b><span style=\"font-weight: 400;\"> It gets arbitrarily close, but for any real x it is still a small positive number.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Skipping the extra step on products.<\/b><span style=\"font-weight: 400;\"> Expressions like x\u00b7e\u207b\u02e3 need L&#8217;H\u00f4pital&#8217;s rule or a growth-rate argument before you can state the limit.<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1b456b7\" data-id=\"1b456b7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5fa9e77 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"5fa9e77\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-61f3118\" data-id=\"61f3118\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-46840bf\" data-id=\"46840bf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a7c3158 elementor-widget elementor-widget-heading\" data-id=\"a7c3158\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Where This Shows Up in Real Life\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3818335 elementor-widget elementor-widget-text-editor\" data-id=\"3818335\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This limit isn&#8217;t just an exam trick. Anything that fades away exponentially relies on it. A radioactive sample follows N(t) = N\u2080e^(\u2212\u03bbt), so the amount left tends to zero as time passes. A cup of tea cools toward room temperature the same way, a drug&#8217;s concentration in the blood drops after a dose, and a capacitor&#8217;s charge dies away as it discharges. In statistics, the tails of the normal and exponential distributions fall off using e\u207b\u02e3-style terms. In each case the fact that e\u207b\u02e3 goes to 0 is what lets us say the effect eventually becomes negligible.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5afab4a\" data-id=\"5afab4a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6461bcb elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"6461bcb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3442d8b\" data-id=\"3442d8b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1e1e5f1\" data-id=\"1e1e5f1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a63a93a elementor-widget elementor-widget-heading\" data-id=\"a63a93a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0185ab5 elementor-widget elementor-widget-text-editor\" data-id=\"0185ab5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The value of e power minus infinity is 0. As the exponent sinks further below zero, e raised to it gets as close to zero as you like without ever turning negative. You can show it by rewriting e\u207b\u02e3 as 1\/e\u02e3, by watching the table of values, or by using the epsilon proof.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8a9b6df\" data-id=\"8a9b6df\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f863ce8 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"f863ce8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c812a69\" data-id=\"c812a69\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9975b17\" data-id=\"9975b17\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-01a8183 elementor-widget elementor-widget-heading\" data-id=\"01a8183\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Frequently Asked Questions\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-af1182a elementor-widget elementor-widget-accordion\" data-id=\"af1182a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1831\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1831\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\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-accordion-title\" tabindex=\"0\">Is e to the power of minus infinity exactly zero?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1831\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1831\"><p><span style=\"font-weight: 400;\">As a limit, yes. For any actual real input, though, the value is a tiny positive number and not exactly zero.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1832\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1832\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\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-accordion-title\" tabindex=\"0\">What is the power of infinity?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1832\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1832\"><p><span style=\"font-weight: 400;\">Infinity, because e\u02e3 grows without bound as x increases.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1833\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1833\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\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-accordion-title\" tabindex=\"0\">What is the power of zero?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1833\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1833\"><p><span style=\"font-weight: 400;\">It&#8217;s 1, since any non-zero number raised to the power 0 equals 1.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1834\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-1834\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\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-accordion-title\" tabindex=\"0\">Can a calculator confirm this?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1834\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-1834\"><p><span style=\"font-weight: 400;\">\u00a0It can support the idea. Try e^(\u221250) and you&#8217;ll see a very small number. Just remember that a calculator shows an example, while the limit is a proof about every large x.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Is e to the power of minus infinity exactly zero?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">As a limit, yes. For any actual real input, though, the value is a tiny positive number and not exactly zero.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What is the power of infinity?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Infinity, because e\\u02e3 grows without bound as x increases.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What is the power of zero?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">It&#8217;s 1, since any non-zero number raised to the power 0 equals 1.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Can a calculator confirm this?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">\\u00a0It can support the idea. Try e^(\\u221250) and you&#8217;ll see a very small number. Just remember that a calculator shows an example, while the limit is a proof about every large x.<\\\/span><\\\/p>\"}}]}<\/script>\n\t\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<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-bb26124\" data-id=\"bb26124\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>What Is the Value of e Power Minus Infinity? 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