{"id":7079,"date":"2016-06-17T16:55:58","date_gmt":"2016-06-17T20:55:58","guid":{"rendered":"https:\/\/mathvault.ca\/?p=7079"},"modified":"2020-07-13T16:28:43","modified_gmt":"2020-07-13T20:28:43","slug":"polynomial-infinity","status":"publish","type":"post","link":"https:\/\/mathvault.ca\/polynomial-infinity\/","title":{"rendered":"The Algebra of Infinite Limits \u2014 and the Behaviors of Polynomials at the Infinities"},"content":{"rendered":"<figure><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7270 aligncenter\" src=\"https:\/\/mathvault.ca\/wp-content\/uploads\/Infinite-Polynomial.jpg\" alt=\"Infinite Limits &amp; Polynomial End-Behavior\" width=\"800\" height=\"480\" title=\"\"><\/figure>\n<p style=\"text-align: justify;\">Regardless of your&nbsp;early surrounding&nbsp;or&nbsp;schooling&nbsp;background, we know for one that there are two kinds of <em>mathematical objects<\/em> that are kind of hard to miss in life. The names? <strong>Polynomial<\/strong> and <a href=\"\/math-glossary\/#infinite\"><strong>infinity<\/strong><\/a>! While the former&nbsp;might have sounded a bit like the name of a <em>snake<\/em>, polynomials is a <em>one-of-its-kind<\/em> mathematical entity whose perfection defies our mathematical imagination.<\/p>\n<p style=\"text-align: justify;\">For one, polynomials are well-known for being&nbsp;<em>infinitely smooth<\/em> and <em>never-ending<\/em>, while at the same time, they could be a <em>line<\/em>, a <em>parabola<\/em>, or any&nbsp;kind of weird, <strong>infinitely-malleable curve<\/strong>&nbsp;ready to&nbsp;assume&nbsp;any shape drawn without <em>lifting<\/em> the pencil (kind of). Heck, polynomials are a&nbsp;favorite object of <em>platonic desire<\/em> among math enthusiasts. Talk about the <em>interaction<\/em> between polynomial and infinity!<\/p>\n<p style=\"text-align: justify;\">So with all that goodness, it makes sense for us to inquire a bit as to why despite of having similar forms, the behaviors of polynomials at the infinities differ, leading to some seemingly unrelated insights about their <strong>properties<\/strong> in general.<\/p>\n<p style=\"text-align: justify;\">All right. Enough said. Time to buckle the seat belt, and let the theoretical musing begins! <!--more--><\/p>\n<div id=\"toc\"><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_1 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Polynomials_%E2%80%94_A_Review\">Polynomials \u2014 A Review<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#A_Venture_Into_the_Infinite_Limits\">A Venture Into the&nbsp;Infinite Limits<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Interactions_Between_Infinity-Converging_and_Constant_Functions\">Interactions Between Infinity-Converging and Constant Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Infinity_Constant\">Infinity + Constant<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Infinity_%C3%97_Constant\">Infinity \u00d7 Constant<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Infinity_Infinity\">Infinity + Infinity<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Infinity_%C3%97_Infinity\">Infinity \u00d7 Infinity<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Behaviors_of_Polynomials_at_the_Infinities\">Behaviors of Polynomials at the Infinities<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Monomials\">Monomials<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#General_Polynomials\">General Polynomials<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Ramifications\">Ramifications<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Afterwords\">Afterwords<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mathvault.ca\/polynomial-infinity\/#Other_Calculus-Related_Guides_You_Might_Be_Interested_In\">Other Calculus-Related Guides You Might Be Interested In<\/a><\/li><\/ul><\/nav><\/div>\n<\/div>\n<h2 id=\"poly\"><span class=\"ez-toc-section\" id=\"Polynomials_%E2%80%94_A_Review\"><\/span><a href=\"#toc\">Polynomials \u2014 A Review<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\">A <em>non-zero<\/em>,&nbsp;real-valued function of the form $\\displaystyle&nbsp;cx^n$ \u2014 where $c \\in \\mathbb{R} \\setminus \\{ 0 \\}$ and $n \\in \\mathbb{N}$ \u2014 forms the building blocks of polynomials. For that reason, they are usually called the (non-zero)<strong> monomials<\/strong>, with&nbsp;the number $n$ being the <strong>degree<\/strong> of the monomial in question. For example, the <em>constant function<\/em> $\\frac{\\pi}{4}$ represents a monomial with degree $0$, whereas the function $ex^{666}$ would be a monomial with degree $666$ (!).<\/p>\n<p style=\"text-align: justify;\">(for the record, the <em>zero function<\/em> also falls into the category of monomials, except that its degree is usually left <em>undefined<\/em>, for the obvious reason that it can be expressed <em>numerous<\/em> forms).<\/p>\n<p style=\"text-align: justify;\">Once there, we can define a <strong>non-zero polynomial<\/strong> as a function of the form:<\/p>\n<p>\\begin{align*} f_1(x)+ \\dots + f_m(x) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">where $m \\in \\mathbb{N_+}$ (i.e., a <em>positive<\/em> integer) and each $f_i(x)$ is&nbsp;a&nbsp;<em>non-zero<\/em> monomial.<\/p>\n<p style=\"text-align: justify;\">In which&nbsp;case, we define the <strong>degree<\/strong> of a <em>non-zero<\/em> polynomial as the <em>maximal<\/em> degree of its monomials (always well-defined, since there&#8217;s only a <em>finite<\/em> number of&nbsp;monomials). For example, the constant function $\\frac{\\ln 2 + 3}{2e}$ represents&nbsp;a polynomial of degree $0$, and&nbsp;the function $3 + 2x^3 + x + x^2$ a polynomial of degree $3$.<\/p>\n<p style=\"text-align: justify;\">(as with before, the <em>zero function<\/em>&nbsp;still falls into the category of polynomials, with the caveat that its degree is usually left undefined \u2014 due to the ambiguity of its leading term)<\/p>\n<p style=\"text-align: justify;\">For the sake of <em>simplicity<\/em> and <em>consistency<\/em> though, we usually prefer to write a <em>non-zero<\/em> polynomial by gathering up the <em>like terms<\/em>, and rearranging the resulting terms so that the monomials are presented in increasing\/decreasing order based on&nbsp;their degrees. Once that&#8217;s done, we refer to the monomial of the highest degree as the <strong>leading term<\/strong> of the polynomial, and its coefficient the <strong>leading coefficient<\/strong> of the polynomial.<\/p>\n<p style=\"text-align: justify;\">For example, instead of writing $10x^3+2x^2+ 5x+ 6x^2 + 5 + 2 x^3$ as our polynomial, we prefer to regroup the like terms and rewrite it as $12x^3 + 8x^2 + 5x + 5$, from which it can be seen&nbsp;that we have a polynomial of <em>degree<\/em> $3$ (i.e., a <strong>cubic polynomial<\/strong>) \u2014 with $12$ as&nbsp;the <em>leading coefficient<\/em>.<\/p>\n<p style=\"text-align: justify;\">In general, a <em>non-zero<\/em> polynomial can have <em>any<\/em> non-zero number as leading coefficient. However, in the special case where the leading coefficient is $1$, it is as if it it&nbsp;kind of disappears out of sight.&nbsp;In which case, we refer to the said polynomial as a <strong>monic polynomial<\/strong>.<\/p>\n<p style=\"text-align: justify;\">In addition, since&nbsp;every <em>non-zero<\/em> polynomial can be re-expressed in terms of<em>&nbsp;monic polynomial<\/em>&nbsp;by <em>factoring out<\/em> the leading coefficient,&nbsp;it sometimes makes sense to&nbsp;analyze the properties of a non-zero polynomial by looking instead at the properties&nbsp;of its <em>underlying&nbsp;monic polynomial<\/em>.<\/p>\n<p style=\"text-align: justify;\">Alternatively, we can also define a polynomial <em>recursively<\/em>&nbsp;as follows:<\/p>\n<div class=\"titlebox orange\">\n<p class=\"titlebox-title\">Definition 1 \u2014 Recursive Definition of Polynomial<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\"><em>A&nbsp;real-valued<\/em> function $p(x)$ is a <strong>polynomial<\/strong> if and only if it satisfies one of the two following options \u2014 through a <em>finite<\/em> number of iterations:<\/p>\n<ol>\n<li style=\"text-align: justify;\">$p(x)=c$ for some $c \\in \\mathbb{R}$ (i.e., a <em>constant function<\/em>).<\/li>\n<li style=\"text-align: justify;\">$p(x) = x q(x) + c$, where $q(x)$ is a <em>polynomial<\/em> and $c \\in \\mathbb{R}$.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">For example, $x^2 + 2x + 1$ is a polynomial by the standard of this&nbsp;<em>recursive definition<\/em>, since $x^2+2x+1 = x ( x+2) + 1$, where $x+2$ is very well a polynomial (why?) and $1$ very well a constant. In general, the recursive definition of polynomial suggests a way of <em>reducing<\/em> a polynomial of degree $n+1$ to that of degree $n$ \u2014 an handy insight when it comes to proving facts about <em>non-zero<\/em> polynomials using the so-called&nbsp;<strong>mathematical induction<\/strong> on the <em>degree<\/em> of polynomials.<\/p>\n<h2 id=\"inflim\"><span class=\"ez-toc-section\" id=\"A_Venture_Into_the_Infinite_Limits\"><\/span><a href=\"#toc\">A Venture Into the&nbsp;Infinite Limits<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\">When tackling the behavior of a function at the infinities, it makes sense that we equip ourselves with some <strong>limit laws<\/strong>&nbsp;on how <em>infinity-converging<\/em> and <em>constant-converging<\/em> functions&nbsp;interact with each other. To that end, we present the following&nbsp;<em>6<\/em> sets of&nbsp;limit laws for your pleasure. \ud83d\ude42<\/p>\n<h3 id=\"constantfunction\"><span class=\"ez-toc-section\" id=\"Interactions_Between_Infinity-Converging_and_Constant_Functions\"><\/span><a href=\"#toc\">Interactions Between Infinity-Converging and Constant Functions<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">To start off, let&#8217;s imagine that we have a function $f(x)$ and a constant function $c$, where $f \\to \\infty$ as $x \\to \\infty$. The question is, what does the function $f(x) + c$ tends to \u2014 if any&nbsp;\u2014 as $x \\to \\infty$?<\/p>\n<p style=\"text-align: justify;\">Here, it shouldn&#8217;t be a surprise that $f(x)+c$ converges to $\\infty$ as well. To see why, we first note that since $f \\to \\infty$ as $x \\to \\infty$, the <strong>definition of limit<\/strong> implies that:<\/p>\n<blockquote><p>Given <em>any<\/em>&nbsp;number \u2014 however <em>large<\/em> \u2014 there&#8217;ll always be <em>some<\/em>&nbsp;<strong>neighborhood<\/strong> $N$ of the form $(\\bigcirc, \\infty)$, such that every member&nbsp;in $f(N)$ <em>exceeds<\/em> this number.<\/p><\/blockquote>\n<p style=\"text-align: justify;\">This means that&nbsp;given <em>any<\/em>&nbsp;number $r$, it would then be&nbsp;possible to find one such&nbsp;<em>neighborhood<\/em> $N$ such that $f(x) &gt; r &#8211; c$ for all $x \\in N$. In which case, we would have that:<\/p>\n<p>\\begin{align*} f(x) + c &gt; &nbsp;r \\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">thereby&nbsp;showing that the function $f(x) + c$ increases&nbsp;<em>beyond bound<\/em>&nbsp;as $x \\to \\infty$ \u2014 regardless of the value of&nbsp;$c$.<\/p>\n<p style=\"text-align: justify;\">Naturally, this leads to the following question: what if $f$ actually converges to $-\\infty$ instead as&nbsp;$x \\to \\infty$? Well, referring to the <em>definition of limit<\/em> again,&nbsp;we have that:<\/p>\n<blockquote><p>Given <em>any<\/em>&nbsp;number \u2014 however <em>negative<\/em>&nbsp;\u2014 there&#8217;ll always be <em>some<\/em>&nbsp;<strong>neighborhood<\/strong> $N$ of the form $(\\bigcirc, \\infty)$, such that every member&nbsp;in $f(N)$ <em>precedes<\/em>&nbsp;this number.<\/p><\/blockquote>\n<p style=\"text-align: justify;\">This means that&nbsp;given <em>any<\/em>&nbsp;number $r$, it would then be&nbsp;possible to find one such&nbsp;<em>neighborhood<\/em> $N$ such that $f(x) &lt; r &#8211; c$ for all $x \\in N$. In which case, we would have that:<\/p>\n<p>\\begin{align*} f(x) + c &lt; r \\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">hence&nbsp;showing that the function $f(x) + c$ decreases&nbsp;<em>beyond bound<\/em>&nbsp;$x \\to \\infty$, and this again regardless of the <em>sign<\/em> and <em>size<\/em> of&nbsp;$c$. Putting everything together, we obtain&nbsp;our <em>first<\/em> set of limit laws concerning <strong>constant functions<\/strong>:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 1 \u2014 Limit Laws Concerning Constant Functions (Sum)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given a <em>real-valued<\/em> function $f(x)$ and a constant function $c$, if as $x \\to \\Box$ (where $\\Box$ could be a number, $+\\infty$ or $-\\infty$), $f(x)$ converges to one of the infinities, then the function&nbsp;$f(x)+c$ converges to the <em>same<\/em> infinity $f(x)$ converges to \u2014 as $x \\to \\Box$.<\/p>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Now, what about the cases where a function is <em>multiplied<\/em> by a constant, as in the case of $c f(x)$ where $c&gt;0$? Well, as logic would have dictated, the <strong>convergence behavior<\/strong> of $c f(x)$ should&nbsp;depend on what $f(x)$ converges to in the first place.<\/p>\n<p style=\"text-align: justify;\">For example, if we know in advance that $f(x)$ converges to $\\infty$ as $x \\to \\infty$, then the definition of limit would imply that given <em>any<\/em> number $r$ \u2014 no matter how&nbsp;<em>positive<\/em>&nbsp;\u2014 we can always&nbsp;find a neighborhood $N$ of the form $(\\bigcirc,\\infty)$ such that:<\/p>\n<p>\\begin{align*} f(x) &amp; &gt; \\frac{r}{c} \\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">That is,<\/p>\n<p>\\begin{align*} cf(x) &amp; &gt; r &nbsp;\\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">showing that the function $cf(x)$ increases <em>without bound<\/em> as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">On the other hand, if $f(x) \\to -\\infty$ instead as $x \\to \\infty$ (remember that $c&gt;0$), then we can similarly infer that given <em>any<\/em> number $r$ \u2014 however&nbsp;<em>negative<\/em> \u2014 it is always possible to find a neighborhood $N$ of the form $(\\bigcirc,\\infty)$ such that:<\/p>\n<p>\\begin{align*} f(x) &amp; &lt; \\frac{r}{c} \\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">That is,<\/p>\n<p>\\begin{align*} cf(x) &amp; &lt; r &nbsp;\\qquad (\\forall x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">showing that the function $cf(x)$ decreases&nbsp;<em>without bound<\/em> as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">Now, this still leaves us with the scenario&nbsp;where $c$ is <em>negative<\/em>. In which case, using very similar&nbsp;arguments, it can still&nbsp;be shown that as $x \\to \\infty$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">$f(x) \\to +\\infty$ implies that $cf(x) \\to -\\infty$.<\/li>\n<li style=\"text-align: justify;\">$f(x) \\to -\\infty$ implies that $cf(x) \\to +\\infty$.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">So that if we combine these insights together, we get our <em>second<\/em> set of limit laws concerning <strong>constant functions<\/strong>:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 2 \u2014 Limit Laws Concerning Constant Functions (Product)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given a <em>real-valued<\/em> function $f(x)$ and a constant function $c$, if as $x \\to \\Box$ (where $\\Box$ could be a number, $+\\infty$ or $-\\infty$), $f(x)$ converges to one of the infinities, then the following cases apply as $x \\to \\Box$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">If $c&gt;0$, then $c f(x)$ converges to the <em>same<\/em> infinity $f(x)$ converges to.<\/li>\n<li style=\"text-align: justify;\">If $c&lt;0$, then $c f(x)$ converges to the <em>opposite<\/em>&nbsp;infinity $f(x)$ converges to.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Interestingly, the above&nbsp;limit laws applies <em>irrespective of the size of $c$<\/em>. For example, $c$ could be $0.00001$ and $f(x)$ could converge to $+\\infty$, and $cf(x)$ would have converged to $+\\infty$ regardless. Likewise, $c$ could be $\\frac{1}{e}$ and $f(x)$ could converge to $-\\infty$, and $cf(x)$ would have converged to $-\\infty$ regardless.<\/p>\n<h3 id=\"infpluscon\"><span class=\"ez-toc-section\" id=\"Infinity_Constant\"><\/span><a href=\"#toc\">Infinity + Constant<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Moving from the constant functions into the general realm,&nbsp;let&#8217;s assume that we&#8217;re given two functions $f(x)$ and $g(x)$, such that as $x \\to \\infty$, $f$ converges to one of the infinities and $g$ converges to some real number $G$. The question is, what can we extrapolate about the <strong>convergence behavior<\/strong> of $f+g$ \u2014 if any?<\/p>\n<p style=\"text-align: justify;\">Here, to figure this&nbsp;out, we begin by noticing that since $g$ converges to an <em>actual<\/em> number $G$, by virtue of the <em>definition of limit<\/em>, there must be&nbsp;a neighborhood $N$ of the form $(\\bigcirc, \\infty)$ such that:<\/p>\n<p>\\begin{align*} G- 1 &lt; g(x) &lt; G+1 \\qquad (\\forall x \\in N)\\end{align*}<\/p>\n<p style=\"text-align: justify;\">In other words, the function $f + g$ has the <em>peculiar<\/em> property that:<\/p>\n<p>\\begin{align*} f(x) + (G- 1) &lt; f(x) + g(x) &lt; f(x) + (G+1) \\qquad (\\forall x \\in N)\\end{align*}<\/p>\n<p style=\"text-align: justify;\">So that if $f(x) \\to \\infty$ as $x \\to \\infty$, then so does&nbsp;the function $f(x) + (G-1)$ (why?). In which case, an application of the&nbsp;<a href=\"https:\/\/en.wikipedia.org\/wiki\/Squeeze_theorem\" target=\"_blank\" rel=\"noopener noreferrer\"><strong>Squeeze Theorem<\/strong><\/a> would show that <em>the same is true<\/em> for the function $f +g$ as well.<\/p>\n<p style=\"text-align: justify;\">In a similar-but-opposite manner, if $f(x) \\to -\\infty$ as $x \\to \\infty$ instead, then the same applies to&nbsp;the function $f(x) + (G+1)$. In which case, invoking the <em>Squeeze Theorem<\/em> again would show that <em>the same is true<\/em> for the function $f + g$ as well, so that if we combine these facts together, we now have a <em>third<\/em> set of&nbsp;limit laws \u2014 this time concerning the <strong>sum of infinity-converging and constant-converging functions<\/strong>:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 3 \u2014 Limit Laws Concerning Infinity-Converging and Constant-Converging Functions (Sum)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given two real-valued functions $f(x)$ and $g(x)$, if&nbsp;as $x \\to \\Box$ (where $\\Box$ could be a number, $-\\infty$ or $-\\infty$), we have both that:<\/p>\n<ol>\n<li style=\"text-align: justify;\">$f$ tends to one of the infinities.<\/li>\n<li style=\"text-align: justify;\">$g$ tends to a real&nbsp;number.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">then the function&nbsp;$f + g$ tends to the <em>same<\/em> infinity $f$ tends to \u2014 as $x \\to \\Box$.<\/p>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Or more concisely,<\/p>\n<p>\\begin{align*} \\infty + c &amp; = \\infty &amp; -\\infty + c &amp; = -\\infty \\end{align*}<\/p>\n<p style=\"text-align: justify;\">where the first identity holds even when $c$ is <em>extremely negative<\/em>, and the second even when $c$ is <em>extremely positive.<\/em><\/p>\n<h3 id=\"inftimescon\" style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Infinity_%C3%97_Constant\"><\/span><a href=\"#toc\">Infinity \u00d7 Constant<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Now, if&nbsp;the&nbsp;two functions $f(x)$ and $g(x)$ are such that as $x \\to \\infty$, $f(x)$ converges to one of the infinities and $g$ converges to some <i>real&nbsp;<\/i>number $G$, what can we say about the <strong>convergence behavior<\/strong> of the function&nbsp;$fg$ \u2014 if any?<\/p>\n<p style=\"text-align: justify;\">Here, it would only seem logical that the convergence behavior of $fg$ depend on the <strong>sign<\/strong> of $G$. For example, if we know in advance that $G$ is <em>positive<\/em>, then the <em>definition of limit<\/em> would dictate that there be a neighborhood $N_1$ of the form $(\\bigcirc, \\infty)$ such that:<\/p>\n<p>\\begin{align*} \\frac{G}{2} = G &#8211; \\frac{G}{2} &lt; g(x) &lt; G + \\frac{G}{2} = \\frac{3G}{2} \\qquad (\\forall x \\in N_1) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">so that if $f(x) \\to \\infty$ as $x \\to \\infty$, then there must be a neighborhood $N_2$ of a similar form where $f(x)&gt;0$. In which case, <em>multiplying&nbsp;the two inequalities<\/em>&nbsp;would show that the function $fg$ satisfies the following&nbsp;<em>peculiar<\/em> property:<\/p>\n<p>\\begin{align*} f(x) \\ \\frac{G}{2} &lt; f(x) g(x) &lt; f(x) \\ \\frac{3G}{2} \\qquad (\\forall x \\in N_1 \\cap N_2) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">Once there, it can be readily seen by&nbsp;<strong>Squeeze Theorem<\/strong> that as $x \\to \\infty$, the function $fg$ converges to the <em>same<\/em> infinity $f(x)$ converges to. Now, that was <em>if<\/em> we know that $f(x) \\to +\\infty$, but if $f(x) \\to -\\infty$ instead as $x \\to \\infty$, then an <em>analogous<\/em> line of reasoning would take us to&nbsp;the same conclusion anyway.<\/p>\n<p style=\"text-align: justify;\">On the other hand, if $G$ were&nbsp;<em>negative<\/em> instead, then by our previous remark, as $x \\to \\infty$, the function $f[-g]$ would converge to the <em>same<\/em> infinity $f$ converges to, which means that <em>by extension<\/em>, as $x \\to \\infty$, the function $fg$ \u2014 which is equivalent to the function $- f [-g]$&nbsp;\u2014 would converge to the <em>opposite<\/em> infinity $f(x)$ converges to. That takes care of the case where $G&lt;0$.<\/p>\n<p style=\"text-align: justify;\">Now, as for the <em>rare<\/em>&nbsp;case where&nbsp;$G=0$, it turns out that there is really not much we can say about the limit of $fg$. In fact, it can be readily seen that as $x \\to \\infty$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">$x \\to \\infty$ and $\\displaystyle \\frac{1}{x} \\to 0$, but the function $\\displaystyle x \\cdot \\frac{1}{x} \\to 1$.<\/li>\n<li style=\"text-align: justify;\">$x \\to \\infty$ and $\\displaystyle \\frac{1}{x^2} \\to 0$, but the function $\\displaystyle x \\cdot \\frac{1}{x^2} \\to 0$.<\/li>\n<li style=\"text-align: justify;\">$x^2 \\to \\infty$ and $\\displaystyle \\frac{1}{x} \\to 0$, but the function $\\displaystyle x^2 \\cdot \\frac{1}{x} \\to \\infty$.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">In other words, the function $fg$ can literally converge to <em>anything<\/em> when $G=0$! And with that last case settled,&nbsp;we can now&nbsp;synthesize all the&nbsp;findings&nbsp;into&nbsp;our <em>fourth<\/em>&nbsp;set of limit laws \u2014 this time concerning the <strong>product&nbsp;of infinity-converging&nbsp;and constant-converging functions<\/strong>:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 4 \u2014 Limit Laws Concerning the Infinity-Converging and Constant-Converging Functions (Product)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given two real-valued functions $f(x)$ and $g(x)$, if&nbsp;as $x \\to \\Box$ (where $\\Box$ could be a number, $+\\infty$ or $-\\infty$),&nbsp;$f$ converges to one of the infinities and $g$ converges to a real number $G$, then the following cases apply as $x \\to \\Box$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">If $G&gt;0$, then the function $fg$ converges to the <em>same<\/em> infinity $f$ converges to.<\/li>\n<li style=\"text-align: justify;\">If $G&lt;0$, then the function $fg$ converges to the <em>opposite<\/em>&nbsp;infinity $f$ converges to.<\/li>\n<li style=\"text-align: justify;\">If $G=0$, then no information can be extracted about the limit of $fg$.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">In other words,<\/p>\n<p>\\begin{align*} \\pm \\infty \\cdot +&nbsp;&amp; = \\pm \\infty &amp; \\pm \\infty \\cdot &#8211;&nbsp;&amp; = \\mp \\infty \\end{align*}<\/p>\n<p style=\"text-align: justify;\">where both identities hold regardless of the <em>magnitude<\/em> of $G$.<\/p>\n<h3 id=\"infplusinf\"><span class=\"ez-toc-section\" id=\"Infinity_Infinity\"><\/span><a href=\"#toc\">Infinity + Infinity<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">When the&nbsp;functions $f(x)$ and $g(x)$ both tends to one of the infinities, interesting <em>interactions<\/em> are bound to happen. For example, if we know that&nbsp;<em>both<\/em> $f$ and $g$ converge to $\\infty$ as $x \\to \\infty$, then the <em>definition of limit<\/em>&nbsp;would guarantee that there be a neighborhood $N$ of the form $(\\bigcirc, \\infty)$ such that:<\/p>\n<p>\\begin{align*} f(x)+g(x) &gt;&nbsp;f(x) + 0 \\qquad (x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">from which we can infer by <strong>Squeeze Theorem<\/strong> that the function $f + g$ converges to $\\infty$ as well \u2014 as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">Similarly, if <em>both<\/em>&nbsp;$f$ and $g$ converge to $-\\infty$ as $x \\to \\infty$, then&nbsp;the <em>definition of limit<\/em> would again guarantee the existence of&nbsp;a neighborhood $N$ of the form $(\\bigcirc, \\infty)$ such that:<\/p>\n<p>\\begin{align*} f(x)+g(x) &lt;&nbsp;f(x) + 0 \\qquad (x \\in N) \\end{align*}<\/p>\n<p style=\"text-align: justify;\">in which&nbsp;case, we can infer \u2014 again by <em>Squeeze Theorem<\/em>&nbsp;\u2014 that&nbsp;the function $f + g$ converges to $-\\infty$ &nbsp;as well \u2014 as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">On the other hand, if $f$ and $g$ converge to <em>opposite infinities<\/em> instead, then it would transpire&nbsp;that there is really not much we can say&nbsp;about the limit of $f + g$, as it can be readily seen that as $x \\to \\infty$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">&nbsp;$\\displaystyle x + 1 \\to +\\infty$ and $\\displaystyle -x \\to -\\infty$, but $\\displaystyle (x + 1) + (-x) \\to 1$.<\/li>\n<li style=\"text-align: justify;\">&nbsp;$\\displaystyle 2x \\to +\\infty$ and $\\displaystyle -x \\to -\\infty$, but $\\displaystyle (2x) + (-x) \\to \\infty$.<\/li>\n<li style=\"text-align: justify;\">&nbsp;$\\displaystyle x&nbsp;\\to +\\infty$ and $\\displaystyle -2x \\to -\\infty$, but $\\displaystyle x&nbsp;+ (-2x) \\to -\\infty$.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">In any case, putting all the findings&nbsp;together, we see that the <em>fifth<\/em> set of limit laws \u2014 this time concerning the <strong>sum of infinity-converging functions<\/strong>&nbsp;\u2014 is now in order:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 5 \u2014 Limit Laws Concerning Infinity-Converging Functions (Sum)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given two real-valued functions $f(x)$ and $g(x)$, if&nbsp;as $x \\to \\Box$ (where $\\Box$ could be a number, $+\\infty$ or $-\\infty$), both $f$ and $g$ converge to the <em>same<\/em> kind of infinity, then the same applies to&nbsp;the function $f + g$.<\/p>\n<p style=\"text-align: justify;\">However, if $f$ and $g$ converge to <em>opposite<\/em> infinities, then no information can be extracted about the limit of $f+g$.<\/p>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Or more concisely,<\/p>\n<p>\\begin{align*} \\infty + \\infty &amp; = \\infty &amp; -\\infty + -\\infty &amp; = -\\infty \\end{align*}<\/p>\n<p>Enjoying those <em>squiggly symbols<\/em> so far? \ud83d\ude42<\/p>\n<h3 id=\"inftimesinf\"><span class=\"ez-toc-section\" id=\"Infinity_%C3%97_Infinity\"><\/span><a href=\"#toc\">Infinity \u00d7 Infinity<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Now, suppose that the functions $f(x)$ and $g(x)$ both converge to one of the infinities, then is it <em>always<\/em> possible to extrapolate something about the limit of the function&nbsp;$fg$? As it turns out, the answer is a resounding <em>yes<\/em>, with the caveat being that this limit varies a bit \u2014 depending on whether $f$ and $g$ converge to the <em>same<\/em> infinity or <em>opposite<\/em> infinities.<\/p>\n<p style=\"text-align: justify;\">For example. if both $f$ and $g$ converge to $\\infty$ as $x \\to \\infty$, then the <em>definition of limit<\/em>&nbsp;would guarantee that there be a neighborhood $N$ of the form $(\\bigcirc, \\infty)$ such that:<\/p>\n<p>\\begin{align*} f(x)g(x) &amp; &gt; f(x) \\cdot 1 \\qquad&nbsp;(\\forall x \\in N)\\end{align*}<\/p>\n<p style=\"text-align: justify;\">from which we can infer by&nbsp;<strong>Squeeze Theorem<\/strong>&nbsp;that the function $fg$&nbsp;<em>increases<\/em> beyond bound as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">And if both $f$ and $g$ converge to $-\\infty$ as $x \\to \\infty$, then our previous finding would again show that $fg$, which is equivalent to $(-f)(-g)$, converges to $\\infty$ all the same \u2014 as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">As for the cases where $f$ and $g$ converge to <em>opposite infinities<\/em>, recycling the <strong>&#8220;negating-the-function&#8221; trick<\/strong>&nbsp;would yield&nbsp;that the function $fg$ now converges to $-\\infty$ instead. For example, if&nbsp;as $x \\to \\infty$, $f \\to +\\infty$ and $g \\to -\\infty$, then we can infer&nbsp;that $f(-g) \\to +\\infty$, which means that&nbsp;the function $fg$ converges&nbsp;to $-\\infty$ as $x \\to \\infty$.<\/p>\n<p style=\"text-align: justify;\">And with all these cases settled, we can now move on into&nbsp;synthesizing the&nbsp;findings into our&nbsp;<em>sixth<\/em> and last set of limit laws \u2014 this time concerning the <strong>product of infinity-converging functions<\/strong>:<\/p>\n<div class=\"titlebox navyblue\">\n<p class=\"titlebox-title\">Proposition 6 \u2014 Limit Laws Concerning Infinity-Converging Functions (Product)<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">Given two real-valued functions $f(x)$ and $g(x)$, if&nbsp;as $x \\to \\Box$ (where $\\Box$ could be a number, $+\\infty$ or $-\\infty$), both $f$ and $g$ converge to one of the infinities,&nbsp;then the following cases apply as $x \\to \\Box$:<\/p>\n<ol>\n<li style=\"text-align: justify;\">If both $f$ and $g$ converge to the <em>same<\/em> infinity, then $fg \\to +\\infty$.<\/li>\n<li style=\"text-align: justify;\">If $f$ and $g$ converge to <em>opposite<\/em>&nbsp;infinities, then $fg \\to -\\infty$.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Or more concisely,<\/p>\n<p>\\begin{align*} \\infty \\cdot \\infty &amp; = \\infty &amp; \\infty \\cdot -\\infty &amp; = -\\infty \\\\ -\\infty \\cdot -\\infty &amp; = \\infty &amp; -\\infty \\cdot \\infty &amp; = -\\infty &nbsp; \\end{align*}<\/p>\n<h2 id=\"endbehavior\"><span class=\"ez-toc-section\" id=\"Behaviors_of_Polynomials_at_the_Infinities\"><\/span><a href=\"#toc\">Behaviors of Polynomials at the Infinities<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\">All right. Now that we have all those limit laws in our <em>bag of tricks<\/em>, we can proceed with confidence into tackling the <strong>end-behaviors<\/strong> of all polynomials: first with the <em>monomials<\/em>, and then the polynomials in general.<\/p>\n<h3 id=\"mono\" style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Monomials\"><\/span><a href=\"#toc\">Monomials<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">For the <strong>constant functions<\/strong> (<em>zero function<\/em> included), the end-behaviors are trivial. As for&nbsp;the monomials with degree $1$ or more, we begin by noticing that:<\/p>\n<ul>\n<li style=\"text-align: justify;\">The function $x$ goes from $-\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">The function $x^2$ goes from $+\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">The function $x^3$ goes from $-\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">$\\ldots$<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">In fact, by using <strong>mathematical induction<\/strong>&nbsp;(coupled with some <em>limit laws on infinities<\/em>), we&nbsp;can show that for all <em>monic<\/em> monomials with degree $1$ or more:<\/p>\n<ul>\n<li style=\"text-align: justify;\">If the monomial&nbsp;has an <em>odd<\/em> degree, then it goes from $-\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">If the monomial&nbsp;has an <em>even<\/em>&nbsp;degree, then it goes from $+\\infty$ to $+\\infty$.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Generalizing a bit further,&nbsp;we&nbsp;see&nbsp;that the end-behaviors of a <em>non-constant monomial<\/em> (i.e.,&nbsp;$cx^n$, where $c \\ne 0$ and $n \\ge 1$) \u2014 in general \u2014 depends on both the&nbsp;<strong>parity<\/strong>&nbsp;of the monomial and the <strong>sign<\/strong> of the&nbsp;coefficient $c$:<\/p>\n<ul>\n<li style=\"text-align: justify;\"><strong>Odd-degree Monomial<\/strong>: If $c&gt;0$, then by the <em>limit law on constant functions<\/em>, $cx^n$ must share the same end-behaviors as $x^n$ \u2014 going from $-\\infty$ to $+\\infty$. On the other hand, if $c&lt;0$, then&nbsp;$cx^n$ goes&nbsp;from $+\\infty$ to $-\\infty$.<\/li>\n<li style=\"text-align: justify;\"><strong>Even-degree Monomial<\/strong>: If $c&gt;0$, then the <em>limit law on constant functions<\/em> again implies that $cx^n$ must share the same end-behaviors as $x^n$ \u2014 now&nbsp;going&nbsp;from $+\\infty$ to $+\\infty$. On the other hand, if $c&lt;0$, then&nbsp;$cx^n$ goes&nbsp;from $-\\infty$ to $-\\infty$.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">And with that,&nbsp;our analysis on the <strong>end-behaviors of monomials<\/strong> is now complete:<\/p>\n<div class=\"titlebox purple\">\n<p class=\"titlebox-title\">Theorem 1 \u2014 End-Behaviors of Monomials<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">The behaviors of&nbsp;a monomial $m(x)$ at the infinities can be analyzed based on its <em>degree<\/em> and the <em>sign<\/em> of its&nbsp;coefficient:<\/p>\n<ul>\n<li style=\"text-align: justify;\">If $m(x)$ is a <strong>constant<\/strong>, then its end-behaviors are trivial.<\/li>\n<li style=\"text-align: justify;\">If $m(x)$ is <strong>odd<\/strong>, then:\n<ul>\n<li style=\"text-align: justify;\">If the coefficient is <em>positive<\/em>, then $m(x)$ goes from $-\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">If the coefficient is <em>negative<\/em>, then $m(x)$ goes from $+\\infty$ to $-\\infty$.<\/li>\n<\/ul>\n<\/li>\n<li style=\"text-align: justify;\">If $m(x)$ is <strong>even<\/strong> (excluding the <em>constant functions<\/em>), &nbsp;then:\n<ul>\n<li style=\"text-align: justify;\">If the coefficient is <em>positive<\/em>, then $m(x)$ goes from $+\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">If the coefficient is <em>negative<\/em>, then $m(x)$ goes from $-\\infty$ to $-\\infty$.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">which \u2014&nbsp;considering that monomials come in all different shapes and forms \u2014 is a pretty amazing achievement. \ud83d\ude42<\/p>\n<h3 id=\"generalpoly\" style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"General_Polynomials\"><\/span><a href=\"#toc\">General Polynomials<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">With the end-behaviors of monomials settled, it&#8217;s only a few steps before we figure out the general end-behaviors of <em>all<\/em> polynomials. Again, similar to the case with&nbsp;monomials, if the polynomial is&nbsp;a <em>constant<\/em>, then its&nbsp;end-behaviors are trivial. As for the&nbsp;other polynomials of the form $\\displaystyle c_n x^n + \\dots + c_0 x^0$ ($n \\ge 1, c_n \\ne 0$), we can factor out the <strong>leading term<\/strong>, thereby producing&nbsp;to the following equality:<\/p>\n<p>\\begin{align*} c_n x^n + \\dots + c_0 x^0 = &nbsp;c_n x^n \\left( 1 + \\dots + \\frac{c_0 x^0}{c_n x^n} \\right)&nbsp;\\qquad (x \\ne 0)&nbsp;\\end{align*}<\/p>\n<p style=\"text-align: justify;\">Once there, leveraging the fact that $\\displaystyle 1 + \\dots + \\frac{c_0 x^0}{c_n x^n}$ tends to $1$ as&nbsp;$x$ tends to $+\\infty$ or $-\\infty$, we can see that $c_n x^n + \\dots + c_0 x^0$ essentially has the same convergence behavior as $c_n x^n$ \u2014 at the<em> infinities<\/em>.&nbsp;That is, the contribution of the <strong>lower terms<\/strong> becomes <em>invariably insignificant<\/em> as we get&nbsp;near the infinities \u2014 an impressive finding which we shall incorporate into&nbsp;our&nbsp;<em>theorem of the day<\/em>!<\/p>\n<div class=\"titlebox purple\">\n<p class=\"titlebox-title\">Theorem 2 \u2014 End-Behaviors of Polynomials<\/p>\n<div class=\"titlebox-content\">\n<p style=\"text-align: justify;\">The behaviors of a&nbsp;polynomial at the infinities can be broken down into the following two cases:<\/p>\n<ol>\n<li style=\"text-align: justify;\">For a <strong>constant polynomial<\/strong>, the&nbsp;end-behaviors are trivial.<\/li>\n<li style=\"text-align: justify;\">Otherwise, it shares the <em>same<\/em> end-behaviors as that of its <strong>leading term<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">Essentially, we&nbsp;get to know about the end-behaviors of <em>any<\/em>&nbsp;polynomial we&nbsp;can come up with, without needing to&nbsp;know how to solve for the&nbsp;<strong>roots<\/strong>, or how the polynomial behaves&nbsp;<em>within<\/em> the infinities! For example, we&nbsp;could be given the function $(x-3)(x-2)(x-1)x(x+1)(x+2)(x+3)$, which has a bit of things going on in between the infinities:<\/p>\n<figure id=\"attachment_7332\" aria-describedby=\"caption-attachment-7332\" style=\"width: 700px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-7332 size-full\" src=\"https:\/\/mathvault.ca\/wp-content\/uploads\/Septic-Polynomial.svg\" alt=\"The Septic Polynomial (x-3)(x-2)(x-1)x(x+1)(x+2)(x+3)\" width=\"700\" title=\"\"><figcaption id=\"caption-attachment-7332\" class=\"wp-caption-text\">The graph of the <strong>septic polynomial<\/strong> $(x-3)(x-2)(x-1)x(x+1)(x+2)(x+3)$<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">However, knowing that we have&nbsp;a polynomial of degree $7$ with <em>positive<\/em> coefficient, we can determine \u2014 with almost zero computation \u2014 that the&nbsp;function essentially travels from $-\\infty$ to $+\\infty$, even without having&nbsp;its graph at our disposal! Now, how is that in for&nbsp;a treat? \ud83d\ude42<\/p>\n<h3 id=\"ram\"><span class=\"ez-toc-section\" id=\"Ramifications\"><\/span><a href=\"#toc\">Ramifications<\/a><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">As can be seen with&nbsp;our earlier finding, an <strong>odd-degree polynomial<\/strong> always goes from one kind of infinity to the other, whereas an <strong>even-degree polynomial<\/strong> (save the <em>constant polynomials<\/em>) stays with the same&nbsp;kind of infinity. This leaves us with <em>two<\/em> kinds of polynomials with fundamentally-different end-behaviors.<\/p>\n<p style=\"text-align: justify;\">In particular, given an <em>odd-degree<\/em> polynomial $p(x)$, the fact that it goes from one infinity to the other suggests that there must be <em>two<\/em> <em>neighborhoods<\/em> \u2014 one of the form $(-\\infty, a]$, and the other of the form $[b, +\\infty)$ \u2014 such $p(x)&gt;0$ in one, and $p(x)&lt;0$ in the other. As a result, we can see that the number $0$ \u2014&nbsp;which is an <em>intermediate value<\/em>&nbsp;\u2014 must have been attained somewhere in the interval $[a,b]$ according to the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Intermediate_value_theorem\" target=\"_blank\" rel=\"noopener noreferrer\"><strong>Intermediate Value Theorem<\/strong><\/a>. Put it simply, an <em>odd-degree<\/em> polynomial always has <em>at least<\/em> one real root \u2014 regardless of the complexity of its shape and form.<\/p>\n<p style=\"text-align: justify;\">In fact, more is true. Using a very similar argument, it can be shown that for every&nbsp;<em>odd-degree polynomial<\/em> has the peculiar property that&nbsp;<em>every<\/em> number \u2014 however large or small \u2014 constitutes an <em>intermediate value<\/em>, and hence must be attained somewhere in the polynomial&#8217;s graph according to the <em>Intermediate Value Theorem<\/em>. Algebraically,&nbsp;this means that&nbsp;the equation $p(x)=c$ always has <em>at least<\/em> one solution \u2014 even if&nbsp;such solutions can be&nbsp;hard&nbsp;to determine&nbsp;or&nbsp;approximate through <em>algebraic\/<\/em><em>numerical<\/em> techniques&nbsp;of&nbsp;<strong>root finding<\/strong>.<\/p>\n<div class='essb-ctt' onclick=\"window.open('https:\/\/twitter.com\/intent\/tweet?text=An+odd-degree+polynomial+is+like+God+%E2%80%94+passing+through+every+single+height+without+leaving+a+trace%21&amp;via=mathvault&amp;related=mathvault&amp;hashtags=math, infinity&amp;url=https:\/\/mathvault.ca\/polynomial-infinity\/', 'essb_share_window', 'height=300,width=500,resizable=1,scrollbars=yes');\">\r\n    \t\t\t<span class='essb-ctt-quote'>\r\n    \t\t\tAn odd-degree polynomial is like God \u2014 passing through every single height without leaving a trace!\r\n    \t\t\t<\/span>\r\n    \t\t\t<span class='essb-ctt-button'><span>Click to Share<\/span><i class='essb_svg_icon_twitter'><svg class=\"essb-svg-icon essb-svg-icon-twitter_x\" aria-hidden=\"true\" role=\"img\" focusable=\"false\" viewBox=\"0 0 24 24\"><path d=\"M18.244 2.25h3.308l-7.227 8.26 8.502 11.24H16.17l-5.214-6.817L4.99 21.75H1.68l7.73-8.835L1.254 2.25H8.08l4.713 6.231zm-1.161 17.52h1.833L7.084 4.126H5.117z\"><\/path><\/svg><\/i>\r\n    \t\t<\/div>\n<p style=\"text-align: justify;\">As an illustration, the <strong>quintic polynomial<\/strong> $-\\frac{\\pi}{3} x^5 &#8211; 2x^4 + 123 x^3 &#8211; x + 5$, which we now know goes from $+\\infty$ to $-\\infty$, must have a <em>root<\/em> somewhere in its graph \u2014 even if we might have very little&nbsp;clue as to how to find it. In fact, what we have here is a&nbsp;polynomial that actually attains <em>every<\/em> single real number in its graph, effectively&nbsp;mapping the set of real numbers&nbsp;<em>to the set of real numbers<\/em>!<\/p>\n<p style=\"text-align: justify;\">Moving on to the other end of the spectrum, an <em>even-degree<\/em> polynomial (excluding&nbsp;the <em>constant polynomials<\/em>), which takes the shape of either a <strong>cap<\/strong>&nbsp;or a <strong>cup<\/strong>, must either attain a <strong>maximum<\/strong>&nbsp;in its graph, or a <strong>minimum<\/strong>&nbsp;in its graph. However, what&#8217;s more interesting is that <em>if<\/em> such a polynomial attains a <em>maximum<\/em>, then any number <em>below<\/em> the maximum represents&nbsp;an <em>intermediate value<\/em>, and hence must be attained somewhere&nbsp;in the graph of the polynomial. Similarly, if the&nbsp;polynomial in question attains a <em>minimum<\/em> instead, then any value <em>beyond<\/em> the minimum constitutes an <em>intermediate value<\/em>, and thus must be attained at some point in the graph of the polynomial as well.<\/p>\n<p style=\"text-align: justify;\">For example, the <strong>quartic polynomial<\/strong>&nbsp;$x^4 &#8211; x -\\frac{1}{2}$, which we know goes from $+\\infty$ to $+\\infty$, must have \u2014 by extension of this reasoning \u2014 attained a <em>minimum<\/em>&nbsp;$m$ somewhere in its graph, for <em>if that&#8217;s not the case<\/em>, then it would be possible to construct a sequence $x_i$ \u2014 which converges to either a real number, $+\\infty$, or $-\\infty$&nbsp;\u2014 such that the sequence $f(x_i)$ converges to $-\\infty$, thereby contradicting the <em>continuity<\/em> and&nbsp;<em>end-behaviors<\/em> of the polynomials in question.<\/p>\n<p style=\"text-align: justify;\">Now, we do want to emphasize that our previous &#8220;proof&#8221; on the existence of the minimum $m$ is neither <em>trivial<\/em> nor <em>constructive<\/em>,&nbsp;in the sense&nbsp;even if we acknowledge the existence of the&nbsp;minimum, it&#8217;s not always possible to find the&nbsp;<em>exact<\/em> location where $m$ is attained through <em>algebraic<\/em>&nbsp;or&nbsp;<em>numerical<\/em>&nbsp;methods. But whatever this&nbsp;$m$ is, what we do know is that <em>each<\/em>&nbsp;value beyond $m$&nbsp;is also attained somewhere in the graph, so that even if we might be clueless as to <em>where<\/em> exactly those&nbsp;values are&nbsp;attained, we are nevertheless&nbsp;correct in asserting&nbsp;that the quartic polynomial&nbsp;$x^4 &#8211; x -\\frac{1}{2}$ maps the set of real number to the interval $[m,\\infty)$.<\/p>\n<p style=\"text-align: justify;\">Bottom line?&nbsp;It seems that regardless of whether&nbsp;the polynomial is <em>even<\/em> or&nbsp;<em>odd<\/em>, there&#8217;s always something we can comment&nbsp;about!<\/p>\n<h2 id=\"afterwords\"><span class=\"ez-toc-section\" id=\"Afterwords\"><\/span><a href=\"#toc\">Afterwords<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\">Wow! That was a fruitful venture with plenty of ideas and insights about&nbsp;<strong>infinite limits<\/strong> and <strong>polynomials<\/strong>! While it&nbsp;definitely leaves a lot&nbsp;more to&nbsp;be said about polynomials and their other exotic properties, the fact that we&#8217;ve mapped out&nbsp;their&nbsp;<strong>end-behaviors<\/strong> and in the process, went through a whole bunch of <strong>limit laws<\/strong> involving the infinities, suggests that we have already covered a lot of ground \u2014 which&nbsp;hopefully represents some&nbsp;intellectual progress as well. \ud83d\ude42<\/p>\n<p style=\"text-align: justify;\">So without further ado, here&#8217;s an <em>interactive table<\/em> summarizing all our findings so far:<\/p>\n<div id=\"wc-shortcodes-tab-1\" class=\"wc-shortcodes-tabs wc-shortcodes-item wc-shortcodes-tabs-layout-box\">\n<ul class=\"wcs-tabs-nav wc-shortcodes-clearfix\">\n<li><a href=\"#\" data-index=\"0\" data-id=\"#wc-shortcodes-tab-polynomial-review\">Polynomial (Review)<\/a><\/li>\n<li><a href=\"#\" data-index=\"1\" data-id=\"#wc-shortcodes-tab-infinite-limits\">Infinite Limits<\/a><\/li>\n<li><a href=\"#\" data-index=\"2\" data-id=\"#wc-shortcodes-tab-polynomial-end-behaviors\">Polynomial (End-Behaviors)<\/a><\/li>\n<\/ul>\n<div class=\"tab-content-wrapper tab-content-hide\">\n<div id=\"wc-shortcodes-tab-polynomial-review\" class=\"tab-content wc-shortcodes-content\">\n<div class=\"wc-shortcodes-row wc-shortcodes-item wc-shortcodes-clearfix\">\n<div class=\"wc-shortcodes-column wc-shortcodes-content wc-shortcodes-one-half wc-shortcodes-column-first \">\n<table>\n<tbody>\n<tr>\n<td><strong>Monomial<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Degree (Monomial)<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Constant Function<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Zero Function<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Non-Zero Polynomial<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Degree (Non-Zero Polynomial)<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"wc-shortcodes-column wc-shortcodes-content wc-shortcodes-one-half wc-shortcodes-column-last \">\n<table>\n<tbody>\n<tr>\n<td><strong>Leading Terms<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Leading Coefficient<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Monic Polynomial<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Polynomial (Recursive Definition)&nbsp;<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Mathematical Induction<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"tab-content-wrapper tab-content-hide\">\n<div id=\"wc-shortcodes-tab-infinite-limits\" class=\"tab-content wc-shortcodes-content\">\n<div class=\"wc-shortcodes-accordion wc-shortcodes-item wc-shortcodes-accordion-collapse wc-shortcodes-accordion-layout-box\" data-behavior=\"autoclose\" data-start-state=\"collapse\">\n<p style=\"text-align: justify;\">Given two <em>real-valued<\/em> functions $f(x)$, $g(x)$ and a constant function $c$, if we take the limit as $x$ tends to $\\Box$ (where $\\Box$ can be either &nbsp;a number, $+\\infty$ or $-\\infty$), then&nbsp;the following 6 sets of&nbsp;<strong>limit laws<\/strong> apply<strong>:<\/strong><\/p>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Constant Functions (Sum)<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If $f(x) \\to +\\infty$, then so does the function $f(x)+c$.&nbsp;<span style=\"line-height: 1.5;\">Similarly, if&nbsp;$f(x) \\to -\\infty$, then so does the function $f(x)+c$.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Constant Functions (Product)<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If $c&gt;0$ and $f(x)$ converges to one of the infinities, then $c f(x)$&nbsp;converges to the <em>same<\/em> infinity $f(x)$&nbsp;converges to.<\/p>\n<p style=\"text-align: justify;\">If $c&lt;0$ and&nbsp;$f(x)$ converges to one of the infinities, then $c f(x)$&nbsp;converges to the <em>opposite<\/em>&nbsp;infinity $f(x)$&nbsp;converges to.<\/p>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Infinity-Converging + Constant-Converging Function<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If $f$ tends to one of the infinities and $g$ tends to a real number, then $f+g$ will tend to the infinity $f$ tends to.<\/p>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Infinity-Converging \u00d7 Constant-Converging Function<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If $f$ tends one of the infinities, and $g$ tends to a real number $G$, then&nbsp;the following cases apply:<\/p>\n<ol>\n<li style=\"text-align: justify;\">If $G&gt;0$, then $fg$&nbsp;converges to the <em>same<\/em> infinity $f$ converges to.<\/li>\n<li style=\"text-align: justify;\">If $G&lt;0$, then $fg$ converges to the <em>opposite<\/em>&nbsp;infinity $f$ converges to.<\/li>\n<li style=\"text-align: justify;\">If $G=0$, then no information can be extracted about the limit of $fg$.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Infinity-Converging Function (Sum)<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If both $f$ and $g$ converge to the <em>same<\/em> kind of infinity, then so does the function $f + g$.<\/p>\n<p style=\"text-align: justify;\">If $f$ and $g$ converge to <em>opposite<\/em> infinities, then no information can be extracted about the limit of $f+g$.<\/p>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Limit Laws \u2014 Infinity-Converging Functions (Product)<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">If both $f$ and $g$ converge to the <em>same<\/em>&nbsp;infinity, then $fg \\to +\\infty$.<\/p>\n<p style=\"text-align: justify;\">If $f$ and $g$ converge to opposite infinities, then $fg \\to -\\infty$.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"tab-content-wrapper tab-content-hide\">\n<div id=\"wc-shortcodes-tab-polynomial-end-behaviors\" class=\"tab-content wc-shortcodes-content\">\n<div class=\"wc-shortcodes-accordion wc-shortcodes-item wc-shortcodes-accordion-collapse wc-shortcodes-accordion-layout-box\" data-behavior=\"autoclose\" data-start-state=\"collapse\">\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">End-Behaviors of Monomials<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">The behaviors of a monomial $m(x)$ at the infinities can be analyzed based on its <em>degree<\/em> and the <em>sign<\/em> of its&nbsp;coefficient:<\/p>\n<ul>\n<li style=\"text-align: justify;\">If $m(x)$ is a <strong>constant<\/strong>, then its end-behaviors are trivial<\/li>\n<li style=\"text-align: justify;\">If $m(x)$ is <strong>odd<\/strong>, then:\n<ul>\n<li style=\"text-align: justify;\">If the coefficient is <em>positive<\/em>, then $m(x)$ goes from $-\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">If the coefficient is <em>negative<\/em>, then $m(x)$ goes from $+\\infty$ to $-\\infty$.<\/li>\n<\/ul>\n<\/li>\n<li style=\"text-align: justify;\">If $m(x)$ is <strong>even<\/strong> (excluding the <em>constant functions<\/em>), then:\n<ul>\n<li style=\"text-align: justify;\">If the coefficient is <em>positive<\/em>, then $m(x)$ goes from $+\\infty$ to $+\\infty$.<\/li>\n<li style=\"text-align: justify;\">If the coefficient is <em>negative<\/em>, then $m(x)$ goes from $-\\infty$ to $-\\infty$.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">End-Behaviors of Polynomials<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">The behaviors of a&nbsp;polynomial at the infinities can be broken down into the following two cases:<\/p>\n<ol>\n<li style=\"text-align: justify;\">For a <strong>constant polynomial<\/strong>, the&nbsp;end-behaviors are trivial.<\/li>\n<li style=\"text-align: justify;\">Otherwise, it shares the <em>same<\/em> end-behaviors as that of its <strong>leading term<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"wc-shortcodes-accordion-trigger \"><a href=\"#\">Ramifications<\/a><\/div>\n<div class=\"wc-shortcodes-accordion-content-wrapper\">\n<div class=\"wc-shortcodes-accordion-content wc-shortcodes-content\">\n<p style=\"text-align: justify;\">A polynomial of <strong>odd degree<\/strong>&nbsp;\u2014 which goes from one infinity to the other \u2014 must attain&nbsp;<em>every<\/em> single real number in its graph. As a result, it maps the set of real numbers to itself,&nbsp;and&nbsp;always have at least a root \u2014 even if such roots&nbsp;could be hard to determine or approximate using <em>algebraic<\/em>\/<em>numerical<\/em> methods.<\/p>\n<p style=\"text-align: justify;\">A&nbsp;polynomial of <strong>even degree<\/strong> \u2014 which stays with the same infinity \u2014 either attains a maximum $M$ (if it takes&nbsp;the shape of a <strong>cap<\/strong>), or a minimum $m$ (if it takes&nbsp;the shape of a <strong>cup<\/strong>). In the first scenario, the polynomial maps the set of real number to $(-\\infty, M]$, and in the second scenario, to $[m, \\infty)$.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"text-align: justify;\">So there you have it! A little venture into polynomials and infinities on the juncture of calculus and real analysis. By the way, do you have a <strong>favorite polynomial<\/strong> of your own? If so, be sure to take a good new look at it using what you&#8217;ve just known!<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Other_Calculus-Related_Guides_You_Might_Be_Interested_In\"><\/span>Other Calculus-Related Guides You Might Be Interested In<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li><a href=\"\/chain-rule-derivative\/\">Chain Rule for Derivative \u2014 The Theory<\/a><\/li>\n<li><a href=\"\/derivative-inverse-functions\/\">Derivative of Inverse Functions: Theory &amp; Applications<\/a><\/li>\n<li><a href=\"\/derivative-tetration-hyperexponentiation\/\">Derivatives of $x^x$, $x^{x^x}$&#8230;<\/a><\/li>\n<li><a href=\"\/exponent-rule-derivative\/\">Exponent Rule for Derivative: Theory &amp; Applications<\/a><\/li>\n<li><a href=\"\/integration-overshooting-method\/\">Integration Series: The Overshooting Method<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>An in-depth exploration of the various limit laws concerning infinities, and the end-behaviors of polynomials at the infinities.<\/p>\n","protected":false},"author":1,"featured_media":10754,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,367],"tags":[],"class_list":["post-7079","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus","category-college-math","post-wrapper","thrv_wrapper"],"_links":{"self":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts\/7079","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/comments?post=7079"}],"version-history":[{"count":0,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts\/7079\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/media\/10754"}],"wp:attachment":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/media?parent=7079"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/categories?post=7079"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/tags?post=7079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}