Find $\lim\limits_{x\to0}\frac{\sin(x)-\tan(x)}{\arcsin(x)-\arctan(x)}$ Find $\displaystyle \lim_{x\to0}\frac{\sin(x)-\tan(x)}{\arcsin(x)-\arctan(x)}$  I tried using L'hopital's rule but it got very messy very fast    UPDATE- So reading about the Taylor series this is what I have so far   $$\lim_{x\to0}\frac{(x-x^3/3!+x^5/5!-\cdots)-(x+x^3/3+2x^5/5+17x^7/315+\cdots)}{(x+x^2/6+3x^5/40+\cdots)-(x-x^3/3+x^5/5-\cdots)}$$  But I'm still stuck
Quadratic Formula in Complex Variables Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers.
I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. I want to prove this:  Prove that if $a \in \mathbb{Z}$ is divisible by $4$, then there exist $b$ and $c$ where $b,c \in \mathbb{Z}$ such that $a = b^2 - c^2$  I want to prove this directily:  $4\mid a $,  $a\in \mathbb{Z} \implies \exists b,c \in \mathbb{Z}, \quad$ s.t. $a=b^2-c^2$  Let $k\in \mathbb{Z}$. Then $4=a\cdot k$ and $k$ can divide $4$. Hence,  $\frac{4}{k} =b^2 - c^2 = (b-c)(b+c) \implies \dfrac{4}{k\cdot(b-c)} = b+c$  Is this proof correct? I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality. I have an idea or subject that should have a Q&amp;A, and want to propose a new site. How can I do this?  Related: Centos mysql version is 5.5 however PHPmyadmin still says version 5.1 Suppose $f$ is a continuous, strictly increasing function defined on a closed interval $[a,b]$ such that $f^{-1}$ is the inverse function of $f$. Prove that,   $$\int_{a}^bf(x)dx+\int_{f(a)}^{f(b)}f^{-1}(x)dx=bf(b)-af(a)$$   A high school student or a Calculus first year student will simply, possibly, apply change of variable technique, then integration by parts and he/she will arrive at the answer without giving much thought into the process. A smarter student would probably compare the integrals with areas and conclude that the equality is immediate.  However, I am an undergrad student of Analysis and I would want to solve the problem "carefully". That is, I wouldn't want to forget my definitions, and the conditions of each technique. For example, while applying change of variables technique, I cannot apply it blindly; I must be prudent enough to realize that the criterion to apply it includes continuous differentiability of a function. Simply with $f$ continuous, I cannot apply change of variables technique.  Is there any method to solve this problem rigorously? One may apply the techniques of integration (by parts, change of variables, etc.) only after proper justification.  The reason I am not adding any work of mine is simply that I could not proceed even one line since I am not given $f$ is differentiable. However, this seems to hold for non-differentiable functions also.  I would really want some help. Pictorial proofs and/or area arguments are invalid. How to customize the ubuntu gnome 3 desktop? This is maybe a stupid question, but how can I choose my own background image on ubuntu gnome 3?? I click on Settings -> Background -> then I click on the current background preview to change it but I see only the defaults images! No + symbol to add or something. On the 'Pictures' tab there is nothing to select, I get an empty white page? I am running ubuntu gnome 13.04 3.9.6-030906-generic.  Thanks! Is a GUID unique 100% of the time? Is a GUID unique 100% of the time?  Will it stay unique over multiple threads? C# HttpWebRequest with XML Structured Data Can't get unicode symbols in math mode I am trying for about a week to launch unicode-math package with XeLaTeX and got nothing. PDF is created well but all Cyrillic symbols in math mode are skipped. I tried using different math fonts - no progress at all. In log file, I found low-level error:  Missing character: There is no [cyrillic letter from input] in font cmmi12!   But all fonts used in document are Unicode ones.  Here is a file I want to be processed well (of course it is UTF-8).  \documentclass[12pt]{book} \usepackage{polyglossia} \setdefaultlanguage[spelling=modern]{russian} \setotherlanguage{english} \defaultfontfeatures{Ligatures={TeX}} \setmainfont{CMU Serif} \setsansfont{CMU Sans Serif} \setmonofont{CMU Typewriter Text}    \usepackage{amsmath, amssymb} \usepackage[russian]{hyperref}  \usepackage{unicode-math} \setmathfont{Latin Modern Math}  \frenchspacing  \begin{document} Просто буквы % Plain letters $$Память: M_{доп}(n) = \Theta(N)$$ % Memory: M_add(n) = \Theta(n) \end{document}   Looking for your help. Is there any easy way to open the document library files in a new window or popup depending on the file types?  e.g. PDF - In Adobe reader (currently opens in IE)  Image file (currently opens in IE) How to access raw texture data in DirectX 11? I want to get access to a texture's pixel color data on the CPU. When trying to map the texture, I get E_INVALIDARG in return from ID3D11DeviceContext::Map.  ID3D11ShaderResourceView* resourceView; //represents my texture, already loaded up //...  //Getting the resources of the texture view... ID3D11Resource* res = nullptr; ID3D11Texture2D* tex = nullptr; resourceView-&gt;GetResource(&amp;res); res-&gt;QueryInterface(&amp;tex); D3D11_TEXTURE2D_DESC desc; tex-&gt;GetDesc(&amp;desc); //Correct data gets filled out D3D11_RESOURCE_DIMENSION dim; res-&gt;GetType(&amp;dim); //value gets set as Texture2D which it should  D3D11_MAPPED_SUBRESOURCE mappedResource; HRESULT h; h = context-&gt;Map(tex,0,D3D11_MAP_READ,0,&amp;mappedResource); //This always fails h = context-&gt;Map(res,0,D3D11_MAP_READ,0,&amp;mappedResource); //This also fails   The idea is that I keep only shaderResourceViews for my textures, because that is all I needed until now, and get the relevant resources here.  I tried mapping as WRITE_DISCARD also, but to no success. I am really confused, because I never had any problem with mapping constant buffers and vertex buffers. I seem to be certainly missing something? By the way, I am using Wic Texture loader and DDS Texture loader for the DirectXTex toolkit, and tried this code for png, dds textures, with and without mip-maps. How to search inside files on Windows 7? In Windows XP we can search for files that contain a defined keyword (inside all files types).  Windows 7 can look inside files for a keyword, but only for text files. (*.doc, *.txt, *.inf, ...), not (*.conf, *.dat, *.*, ...).   don't contain any filter I can use for this.  How is this possible? I am using cURL to insert data into salesforce. It works with single record but I need to insert multiple records at a time. Is it possible to do so and is it possible via a REST interface? Flight cancelled but doesn't show up on status I booked a SpiceJet flight from Dubai to Ahmedabad on Dec 16 (SG-16) scheduled to depart at 7:25 PM. At noon I received an SMS from SpiceJet to advise me that my flight has been delayed by 4h 50m. Sure enough, when I checked flight status on SpiceJet's website, that was the case. But on other websites like flightstats.com and dubaiairports.ae, the status was that flight was "delayed by 2h 25m".    At 5 PM, I got an SMS that flight had been cancelled. I double checked on Spicejet's website and that too showed the flight to be cancelled. But both flightstats.com and dubaiairports.ae still showed the flight as delayed by 2h 25m.    I went to the airport and met SpiceJet officials. Sure enough, the flight was cancelled. But somehow, the flight cancellation status never made it to the system. I have a Tripit Pro account, and even that never sent me an alert for that. I contacted the travel agent through which I booked my flight (makemytrip) for a refund, and they said that their system doesn't show that flight was cancelled. Now, even the Spicejet website has removed the cancellation status from that flight.    Apart from the SMS, I now have no proof that my flight was cancelled.    Can someone tell me, how can I proceed with a refund claim? If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. How to get full query text from performance_schema.threads? According to : Access to the threads table does not require a mutex  and has minimal impact on server  performance. The other sources have negative performance consequences because they require a mutex.   So yes, I want to use threads table to examine running queries. The trouble is, the PROCESSLIST_INFO column does not contain the full query. It just contains a truncated version of the query. There is a flag viz performance_schema_max_sql_text_length which controls how many bytes are allocated for the SQL query. But it seems to affect the events_statements_history_long table. There is no change in the thread table output. There is also the events_statement_current table. But it is not clear if accessing that table affects performance or not. What kinds of compensation can I get from Deutsche Bahn for canceled and late trains? I can never remember how I do this because it comes up so infrequently for me. But in C or C++, what is the best way to read a character from standard input without waiting for a newline (press enter).  Also ideally it wouldn't echo the input character to the screen. I just want to capture keystrokes with out effecting the console screen. Can Java's NullPointerException be changed to report on which variable was null?