The problem:  $$g(x) = \frac{x^3-8}{x-2}$$ Explain why $g$ is not defined at $x=2$.  My solution: $$g(x) = \frac{x^3-8}{x-2} = \frac{(x-2)(x^2+2x+4)}{x-2}$$ The two $(x-2)$ get canceled $\implies x^2 + 2x + 4$. --&gt; This should be undefined at $x=2$, not really. But if we don't change the function it is of course not defined at $x = 2$, because it would give us a $0$ (zero) at the denominator. That gives us infinity. Please help, not if sure if I am on the right track. Thank you very much
$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much.
Since reading the answers to my question about  I've been good about giving my meat time to defrost in the refrigerator. However, my plans for tonight have changed and I'd like to defrost some ground beef that is completely frozen for dinner this evening. What are my options to do so quickly and safely? How to force nginx to resolve DNS (of a dynamic hostname) everytime when doing proxy_pass? What Easter eggs do the chat sites have? According to , the SE chat sites may have individual Easter eggs, depending on the site. If you find them, please post. Because it's so easy to make this stuff up, a screenshot as proof would be nice. (Although Photoshop isn't that hard either...) Is a GUID unique 100% of the time? MySQL Hashing Function Implementation I know that php has md5(), sha1(), and the hash() functions, but I want to create a hash using the MySQL PASSWORD() function.  So far, the only way I can think of is to just query the server, but I want a function (preferably in php or Perl) that will do the same thing without querying MySQL at all.  For example:  MySQL hash -> 464bb2cb3cf18b66  MySQL5 hash -> *01D01F5CA7CA8BA771E03F4AC55EC73C11EFA229  Thanks! Newbe Ubuntu installer - Ubuntu 14.04.3 LTS 64bit - Windows 10 Having some issues installing Ubuntu 14.04.3 LTS 64bit next to Windows 10.  This is what I did:  1) Created a bootable usb with ubuntu on it using Universal-USB-Installer-1.9.6.3  2) Changed the boot mode  3) Boot from USB - It showed the option to test drive or install - I chose install  4) Choose Language  5) Setup Wireless - I didn't choose anything here  6) Then it showed me the preparing to install  7) After that a message popped up saying:      Unmount partitions that are in use The installer has detected that the following discs have mounted partitions /dev/scd Do you want the   installer to try and unmount these partitions before continuing? If   you leave them mounted, you will not be able to create, delete, resize   partitions on these discs, but you may be able to install on an   existing partition there.   So obviously I said yes.  8) Installation type - I chose something else - This looked nothing like the screen shots I saw on the installation guide.    9) Then it only gave me the option to choose my bootable usb for the installation. With a bunch of options to choose, so I got scared and quit.     Can any one please help?  Keep in my mind I do not know a lot, so please dumb it down. Why can't I choose to run it along side Windows 10? Why doesn't the installation pick up my hard drives?  If you need any other info, please ask. But I think I put everything in here. Pre-1980s Sci-fi movie about killer aliens Why are there no animals in "The Matrix"? pdfx: incomplete CIDset in some fonts I try to compile a document as pdf/x and pdf/a with pdfx under LuaLaTeX, and I am already quite successful. One thing still poses problems though: With some fonts I get the following error if I check the resulting pdf/a file with Acrobat: "CIDset in subset font is incomplete (font contains glyphs that are not listed)" (VeraPDF gives the same error, so the issue seems to be real).  Here's a MWE:  \RequirePackage{filecontents} \begin{filecontents*}{\jobname.xmpdata} \Title{Test}       \Author{Simon Spiegel}       \Language{de-DE} \end{filecontents*} \documentclass {article} \usepackage{fontspec} \setmainfont{Palatino Linotype} [Ligatures={TeX,Common}, Extension = .ttf, ] \usepackage[a-3b]{pdfx} \begin{document} Test \end{document}   I realise that not everyone will be able to compile this MWE due to the proprietary font, but since the problem only occurs with specific fonts, I had to use this example.  The pdfx manual mentions this error, but only in relation to XeLaTeX. According to the manual, it should not occur with LuaLaTeX.  From what I understand, this seems to be at least to some degree a problem of this specific font, but since I am relying on it, I am looking for a way to fix it.  EDIT  Here's a new MWE with a bunch of fonts which are hopefully more common (the paths have to be adapted of course).  \RequirePackage{filecontents} \begin{filecontents*}{\jobname.xmpdata} \Title{Test}       \Author{Simon Spiegel}       \Language{de-DE} \end{filecontents*} \documentclass {article} \usepackage{fontspec} \setmainfont{Georgia} [Ligatures={TeX,Common}, Extension = .ttf, ]  \newfontfamily\fontone{Verdana} [Ligatures={TeX,Common}, Extension = .ttf, ] \newfontfamily\fonttwo{Cambria} [Ligatures={TeX,Common}, Extension = .ttf, Path = {/Users/simi/FontExplorerX/C/Cambria/} ] \usepackage[a-3b]{pdfx} \begin{document} Test  \fontone Test  \fonttwo Test \end{document}   All these fonts give me the same error. I wonder if this only happens with .ttf OpenType fonts. $a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$  Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$  If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it. Is there a BNF grammar of the TeX language? I'm looking for a BNF grammar of the TeX language, does it exist?  EDIT  For those of us who are not computer scientists, a BNF grammar is one kind of formal description of a CFG: .  For those who don't know their Chomsky, a CFG is a , which means (very roughly) that substitutions can be made independently of their use context. I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase? Does this sentence seem weird? "Goose"–"geese" vs. "moose"–"moose" Why is it that the plural of goose is geese but the plural of moose is moose? The same goes for mouse and house. Mouse becomes mice, yet house becomes houses. In English, words with a 'g' followed by a front vowel (e, i, y) can be pronounced with a soft g or a hard g:   Words with Germanic roots are usually pronounced with a hard g: gear, get, gift, give Words with Latin and Greek roots are usually pronounced with a soft g: gem, general, giraffe, giant   But how should a purely English word (if such thing even exists) with a 'g' followed by a front vowel be pronounced? In other words, if an English speaker saw a new word of unknown origin (eg: a neologism) that starts with gi- or ge-, how would they pronounce it? I need to repair my Windows 10 installation. Where can I get a clean ISO (with install.wim, no Install.ESD, so that I can run DISM against the WIM) of the specific Windows 10 build required for this?  In my case, I need the Anniversary update (Build 14393), Creators Update (Build 15063) or Fall Creators Update (Build 16299.15). Why are high white balance temperatures redder when warmer objects are bluer? I know it's dumb to ask this type of questions. But, I still don't get this. I've read the "Black body" story and know that when it's heated, it gives off colors; 'warm' colors (red→orange→yellow..) that have low color temperatures, at low temperatures, (I mean, when it is cooler than afterwards) right? And, the black body gives off 'cold' colors (white→blue..) that have high color temperatures at high temperatures (when it is hotter than before), right? Correct? So there're two, yes, very dumb, questions and the latter one is dumber..     Why are that 'warm' colors said to have low temperatures? And 'cold' colors vice versa? So the redder colors have lower c/t and bluer colors have higher c/t, then   why does the camera's color temperature - adjusting slider has the bluer colors on the left, and the redders on the right?       Please, forgive me for my ignorance... If neutrinos are disfavoured as DM candidates why aren't axions? Numerical simulations of observed large-scale structure formation work best with Cold Dark Matter (CDM; see the answer ). Neutrinos are candidates for Hot Dark Matter (HDM), and hence they cannot account for the total dark matter abundance of the Universe. By the same token, axions are also relativistic because they have very tiny masses. Aren't they also candidates of HDM like neutrinos? Shouldn't they also be disfavoured for the same reason? Removing nodes from an XmlDocument The following code should find the appropriate project tag and remove it from the XmlDocument, however when I test it, it says:  The node to be removed is not a child of this node.  Does anyone know the proper way to do this?  public void DeleteProject (string projectName) {     string ccConfigPath = ConfigurationManager.AppSettings["ConfigPath"];      XmlDocument configDoc = new XmlDocument();      configDoc.Load(ccConfigPath);      XmlNodeList projectNodes = configDoc.GetElementsByTagName("project");      for (int i = 0; i &lt; projectNodes.Count; i++)     {         if (projectNodes[i].Attributes["name"] != null)         {             if (projectName == projectNodes[i].Attributes["name"].InnerText)             {                                                                 configDoc.RemoveChild(projectNodes[i]);                 configDoc.Save(ccConfigPath);             }         }     } }   UPDATE  Fixed. I did two things:  XmlNode project = configDoc.SelectSingleNode("//project[@name='" + projectName + "']");   Replaced the For loop with an XPath query, which wasn't for fixing it, just because it was a better approach.  The actual fix was:  project.ParentNode.RemoveChild(project);   Thanks Pat and Chuck for this suggestion. I can't destroy blocks in Minecraft Creative Mode (singleplayer)