A conceptual doubt in limits
Why is $1^{\infty}$ considered to be an indeterminate form
Elementary properties of closure Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks.  Definition: Let $X\subset \mathbb{R}$ and let $x'\in \mathbb{R}$, we say that $x'$ is an adherent point of $X$ iff $\,\forall\varepsilon&gt;0\,\exists x\in X \text{ s.t.} \; d(x',x)\le \varepsilon$. We say that $\overline{X}$ is the closure of $X$ iff contain all the adherent points of $X$.  Lemma (Elementary properties of Closures): Let $X$ and $Y$ be arbitrary subset of $\mathbb{R}$. Then $X\subset \overline{X}, \; \overline{X\cup Y}=\overline{X}\cup\overline{Y},\; \overline{X\cap Y}\subset\overline{X}\cap\overline{Y}$. If $X\subset Y$ then $\overline{X}\subset \overline{Y}$.  Proof:  Clearly if $x\in X$,  $x$ is $\varepsilon$-adherent to $x$ (indeed is $0$-close to itself) for every $\varepsilon$, hence $x\in \overline X$.   Let $z'$ be an adherent point of $X\cup Y$, then there exists some $z\in X\cup Y$ such that $d(z',z)$ for a given $\varepsilon&gt;0$. Suppose  $z\in X$, hence $z'$ is an adherent point of $X$, i.e., $z'\in \overline{X}\subset \overline{X}\cup \overline{Y}$. Similarly when  $z\in Y$, $z'\in \overline{Y}\subset \overline{X}\cup \overline{Y}$. Thus $ \overline{X\cup Y}\subset\overline{X}\cup \overline{Y}$. We  know that $X\cup Y \subset\overline{X\cup Y}\subset\overline{X}\cup \overline{Y}$. Then $\overline{\overline{X\cup Y}}=\overline{X}\cup \overline{Y}$. Since $\overline{\overline{X\cup Y}} = \overline{X\cup Y}$ we're done.  Let $z'\in \overline{X\cap Y}$, i.e., $z'$ is an adherent point of $X\cap Y$. So, there is some $z\in X\cap Y \subset X$ such that $d(z',z)\le \varepsilon$ for any given $\varepsilon&gt;0$. It follows that $z'$ is an adherent point of $X$, i.e., $z'\in \overline{X}$. Similarly since $z\in X\cap Y \subset Y$ we have $z'\in \overline{Y}$. Hence $z'\in \overline{X}\cap\overline{Y}$.   Let $x'\in \overline{X}$ we'd like to show that $x'\in \overline{Y}$. Let $\varepsilon&gt;0$ be arbitrary then $\exists x\in X\subset Y$ such that $d(x',x)\le \varepsilon$, so $x'$ is an adherent point of $Y$, i.e., $x'\in \overline{Y}$ as desired. $\Box$     Claim 1: If $X\subset Y \subset \overline{X}$. Then $\overline{Y}=\overline{X}$.    Proof claim: Let $y'$ be an adherent point of $Y$, i.e., $y'\in \overline{Y}$ and let $\varepsilon&gt;0$ be given. Thus, $\exists y \in Y\subset \overline{X}$ s.t. $d(y,y')\le \varepsilon$. So, $y$ is an adherent point of $X$. Thus $\exists x\in X$ s.t. $d(x,y)\le \varepsilon$ and therefore  we must have $d(x,y')\le 2\varepsilon$ which shows that $y'$ is an adherent point of $X$, i.e., $\overline{Y}\subset \overline{X}$. Conversely if $x'$ is an adherent point of $X$. Then $\exists x\in X\subset Y$ s.t. $d(x,x')\le \varepsilon$, and so is adherent to $Y$. $\Box$     Claim 2: $\overline{\overline{X}} = \overline{X}$    Proof: It will suffice to show that if $x''\in \overline{\overline{X}}$, then $x''$ is an adherent point of $X$. Let $\varepsilon &gt;0$ be arbitrary and let $x''$ be an adherent point of $\overline{X}$. Then there is some $x' \in \overline{X}$ such that $d(x'',x')\le \varepsilon$. Since $x' \in \overline{X}$ by definition is an adherent point of $X$, so there is some $x\in X$ for which $d(x,x')\le \varepsilon$. Thus $d(x,x'')\le d(x,x')+d(x',x'')\le 2\varepsilon$ it follows that $x''$ is an adherent point of $X$ as desired. $\Box$ Can I get a rebate after using my HSA? How much sleep do adults actually need? The common consensus is that most adults should get 8 hours of sleep a night.  However, sleep needs are highly variable, with some people needing more to function properly, and some gaining little benefit past just a few hours. I'm trying to learn more about the statistics here.  I'd imagine sleep needs would fit a normal distribution, with a median of around 8 hours and a standard deviation of around an hour. The  would then put 68% of adults in the 7-9 hour range, 95% in the 6-10 hour range, and 99.7% in the 5-11 hour range, but that's just a guess.  Has this distribution been studied, and where? How to delete a keyboard layout in Windows 10 I installed Windows 10 on a physically German keyboard, then installed a Window English software keyboard and bought some little stickers to make my keyboard look OK.  However, Windows shows 2 keyboards installed &amp; occasionally toggles back to German.  Note that I am not talking about removing a language (which is all that I can find when I Google).  How do I remove the German keyboard layout? How to avoid bad box with long words with no hyphenation? How can I convert String to Int? Humans don't digest fibre, and nor do any animals, but some creatures have micro-organisms (bacteria, I'm guessing) in their gut which digest fibre for them.  My question is, could humans take some sort of pill and establish our own fibre-digesting colonies in our intestines?  Has this been tried?  Supposing it could be done, would you suppose it would have any negative effects on our health?  (i.e.. would those whose gut digested fibre for them be stuck on a low-fibre diet?).    Purpose of question: just wondering.  It would be cool to be able to eat bark. Object emit light distance I created an object, and I wanted it to glow. I used to just place points of light around it, and parent them to it, but using the emit option in the materials seemed so much easier.  But when I gave it an a higher emit than 0, it went only a small distance before the light stopped. I turned it up to 200, and the object was white with light, but the light didn't go any farther. Oh and I changed the color, and nothing happened. If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited I want to put a table and a figure side-by-side. The table should fill two third of the text width, and the figure the last third.  So I'd like to achieve one of the following effects (which are equivalent in my case):   One table with 3 columns. The first two columns contain text, the third column contains a lengthy image. The height of the image should be determined by the height of the whole table, if that's possible Or, a divided page like in newspapers. On the left side I have the table, on the right side the image.   What I have tried:  \begin{multicols}{2} \begin{tabular}{ l l } Text &amp; Text \\ Text &amp; Text \\ ... &amp; ... \\ \end{tabular} \vfill \begin{figure}[H]     \includegraphics[]{images/image} \end{figure} \end{multicols}   But this places the picture directly on top of the table.  I also tried using \usepackage[export]{adjustbox} and \includegraphics[right]{images/image} but the caption is still in the center of the page and therefor on top of the table. What does the explicit keyword mean? Determining if trees within forest gaps are clustered using R? I use the native Clock app, and my alarm used to just have a normal ringtone. I assume there was some automatic update recently because this morning, in addition to the normal ringtone, there was a robotic voice saying "It is 7:25 AM." Then later I got a notification from the native Calendar app, and in addition to my normal ringtone, the robotic voice announced the text of the notification.  How do I shut this off? It's really annoying and I don't see an option in Settings. Is there any function that converts an escaped Url string to its unescaped form? System.Web.HttpUtility.UrlDecode() can do that job but I don't want to add a reference to System.Web.dll. Since my app is not a web application, I don't want to add a dependency for only using a function in an assembly.  UPDATE: Check  about the same issue. How can I iterate over an enum? Let $a$ and $b$ be natural numbers with $ab&gt;2$. Suppose that the sum of their highest common factor and least common multiple is divisible by $a+b$. Prove that the quotient is at most $\frac{a+b}4$. When is this quotient exactly equal to $\frac{a+b}4$. Mathematica envelope for the bottom of a plot, a generic function Average run length Suppose in an Xbar chart, k successive points plotted outside a control limit indicates an out of control situation. If p is the probability of plotting a point outside a control limit, how is ARL = $(1-p^k)/(p^k (1-p))$? The temperature of an electron Does an electron have a temperature, if so, what is it?  Imagine an electron (Ke = 1 eV) in a tube at room temperature (300 K)    what is its temperature?   Imagine now same electron in space (3 K) with same Ke, is it any different from the other?  What is the influence of external temperature on electrons?  Do those electrons have electric energy, or do we have such energy only when the electron hits something and discharges its Ke? Why is "glide symmetry" its own type? Artin's Algebra pages 155 &amp; 156 list the types of symmetry of a plane figure as:   Reflective Rotational Translational Glide   He then goes on to say "Figures such as wallpaper patterns may have two talk about other figures having combinations of independent" symmetries. EDIT: He says "... having combinations of independent translational symmetries". See Joriki's answer.  Why doesn't glide symmetry count as having two independent (reflection + translation) symmetries? If we're going to count combinations, why not have rotation + translation etc. as their own types too?