How    $\lim_{h\to 0} \sin h$ when divided by $h$ gives the value $1$ ? Does it also follows with other ratios that for example. $\lim_{h\to 0} \cos x/x = 1$
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.
I am looking for a driver for the TP-Link AC600 Archer T2U Nano, on Kubuntu 18.04 (kernel version 4.15.0-51-generic).   I already tried different drivers, none of them worked. Why don't you have to press CTRL+ALT+DEL to logon anymore in Windows 8? In previous versions of Windows, you had to press Ctrl + Alt + Delete to logon, or to unlock a locked workstation.  This was because this key sequence was recognized only by the OS and thus other software couldn't intercept it and display a spoofed logon screen to capture passwords.  Starting with Windows 8, you now just have to press Enter to get to the logon screen.  What is to stop someone from writing a fake logon screen?  Did Windows 8 add some sort of new security mechanism to mitigate this security issue? How can I get a significant overall ANOVA but no significant pairwise differences with Tukey's procedure? distance between a line and a point vector formula How do I install additional drivers? I've just installed Ubuntu and I need either video, wireless, or some sort of other manufacturer (non-opensource) driver. How can I run the tool that will fetch all of these for me? I'd like graphical instructions.  Related: If everything in the universe doubled in size overnight, would it be noticeable? Read integers from external data file to document programming code in LaTeX I am attempting to document my MATLAB programming code in LaTeX. For  this, I use the mcode package, which defines the MATLAB lstlisting  environment. With the commands  \newcommand{\mfileMain}{main.m} \newcommand{\mfileMainFirstNumber}{\mfileMainFirstLine} \newcommand{\mfileMainFirstLine}{26} \newcommand{\mfileMainLastLine}{33} \lstinputlisting[firstnumber=\mfileMainFirstNumber,                  firstline=\mfileMainFirstLine,                  lastline=\mfileMainLastLine]{\mfileMain}   I set the firstnumber, firstline and lastline parameters and use  them to import a piece of MATLAB code into the LaTeX document with  \lstinputlisting. However, when I add some code before the  \mfileMainFirstLine, in this case 26, the code is shifted and the  documentation is ruined. To fix this, I would have to update the line  numbers.  I am trying to fix this issue by adding 'tracers' to my MATLAB  program. I use the commands  filename=['lstinputlisting_line_numbers_' mfilename '.txt']; fileID=fopen(filename,'w');   to create a text file in which I store the line numbers for which the  code is documented using the command  dbstack_eval=dbstack;fprintf(fileID,[num2str(dbstack_eval.line+1) '\n']);   . This results in a text file lstinputlisting_line_numbers_.txt with  content for instance like  53 62 75 91   in case I want to document four parts of the MATLAB program main.m.  What I want to do from here is simply use this line number data in the  \lstinputlisting commands. Perhaps it is possible to store the line  numbers in the \newcommands{}{}, or perhaps there is a better, more  elegant way of doing it. In this way, the line numbers would update  automatically in case I add new lines of code. Consequently, my LaTeX  document would still be okay, saving me a lot of time not having to  update the line numbers myself.  A minimum working example could look like the following. The LaTeX file  main.tex would have the content  \documentclass[a4paper]{article} \usepackage[framed,numbered]{mcode}  % Data for the 'lstinputlisting' environments. \newcommand{\mfileMain}{main.m} \newcommand{\mfileMainFirstNumberA}{\mfileMainFirstLineA} \newcommand{\mfileMainFirstLineA}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainLastLineA}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainFirstNumberB}{\mfileMainFirstLineB} \newcommand{\mfileMainFirstLineB}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainLastLineB}{WHAT SHOULD I PUT HERE?}  \begin{document}     In the code     \lstinputlisting[firstnumber=\mfileMainFirstNumberA,                      firstline=\mfileMainFirstLineA,                      lastline=\mfileMainLastLineA]{\mfileMain}     we calculate the ....       And in the code     \lstinputlisting[firstnumber=\mfileMainFirstNumberB,                      firstline=\mfileMainFirstLineB,                      lastline=\mfileMainLastLineB]{\mfileMain}     we calculate the..... \end{document}   . The MATLAB file main.m would have the content  %% Line number output.  filename=['lstinputlisting_line_numbers_' mfilename '.txt']; fileID=fopen(filename,'w');  dbstack_eval=dbstack;fprintf(fileID,[num2str(dbstack_eval.line+1) '\n']);     %% Input to be specified by user.  alpha=10; % PDE parameter. x_s=0; % Starting coordinate. x_e=1; % End coordinate. N_n=1e2+1; % (N)umber of (n)odes. N_ip=2; % (N)umber of (i)ntegration (p)oints.  dbstack_eval=dbstack;fprintf(fileID,[num2str(dbstack_eval.line+1) '\n']);     fclose(fileID); % Close the file 'fileID'. Fell for fake sugar daddy. How do I get out? I fell on hard times and for the first time I attempted to get a sugar daddy. He asked for my bank account login info and reluctantly I gave it to him even though I knew I shouldn't have. Short story, I figured out he's fake and called him out on it after he made a couple of mobile deposits. Every time he made a mobile deposit he would go in there and take a third of the money he deposited. I also noticed he somehow used an old check I deposited and changed it and used it again. I got him to admit he wasn't who he claimed to be and I changed my login info. He claimed he could still see how much money was in my account and sent me a screenshot of how much money was in the account with the amount and &quot;Chase checking&quot; before it. That was the only information in the screenshot. He also threatened to get me in trouble with the law if I didn't send him more money. I changed my login info again just in case but I could really use some advice on what should I do next. Notation problem in integration: write $dx$ or ${\mathrm{d}}x$? How do I work recursively through a directory tree and execute a specific command on each file, and output the path, filename, extension, filesize and some other specific text to a single file in bash. Why does aptitude give me intel-microcode upgrades when I have an AMD processor? I have a Supermicro AS-2022G-URF (H8DGU motherboard) with 2 AMD Opteron 6272 16C/16T 2.1GHz and an LSI MegaRAID SAS 2108 RAID controller currently running Ubuntu Server 14.04 LTS (scheduled to be upgraded to 16.04 this summer). Several times over the last year, when I have done aptitude update/upgrade, the intel-microcode package has shown up as one of the packages that will be installed. Since this server has AMD processors, not Intel, Why does aptitude offer this package? Is it in any way necessary for other reasons? Any ill effects from going along with aptitude on this? 7 segment binary to hex How to move and overwrite subdirectories (and files) to parent directory? Vertical space in lists How to use a variable for a key in a JavaScript object literal? An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Where $(50)_5 = 50! / 45! $ and $50\choose 5$ $= 50!/(5!45!)$   I've narrowed it down to these two possibilities (though even these may be wrong), but I think I'm confusing myself as to which one I use. Can anyone suggest a good way of thinking of this problem that will make it apparent whether I need use use the falling factorial or the binary coefficient?   Edit: thank you for all the answers! Indeed, I meant $50!/45$ as the falling factorial. Also, the question doesn't specify whether it was meant to be ordered or unordered, but I think I understand my confusion now! Basic mathematical induction regarding inequalities Proof of the First Isomorphism Theorem for Groups The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit worried about the (*) line. It looks a bit awkward, but is it too bad? Any suggestions?    Theorem: If $\varphi : G \rightarrow H$ is a homomorphism of groups, then $ker \varphi \trianglelefteq G$ and $G/ker \varphi \cong \varphi (G)$.  Proof: Let $\varphi:G \rightarrow H$ be a group homomorphism. If $g \in G$ and $h \in ker \varphi$, then we have that $ghg^{-1} \in ker \varphi$ because $$\varphi (ghg^{-1})= \varphi (g) \varphi (h) \varphi (g^{-1})= \varphi (g) \varphi (g^{-1}) = \varphi (gg^{-1}) = \varphi (1) =1,$$ so that $ker \varphi \trianglelefteq G$. Now, define $\phi: G/ker \varphi \rightarrow \varphi (G)$ by $$\phi (g(ker \varphi))=\varphi(g),$$ for each $g(ker \varphi) \in G/ker \varphi$, for some $g \in G$. This is well defined because if $g' \in g(ker \varphi),$ then $$\phi (g'(ker \varphi))= \varphi(g')= \varphi(g)= \phi (g(ker \varphi)).$$ Also, this is a group isomorphism because for each $\varphi (h) \in \varphi (G),$ for some $h \in G$, we have $$(*) \ \phi^{-1}\{\varphi(h)\}= \phi^{-1} \varphi[g(ker \varphi)]= \phi^{-1} \varphi(\pi^{-1}\{g(ker \varphi)\})= \{g(ker \varphi)\},$$ a set with a single element of $G/ker \varphi$, so that it is a bijection, and for every $g(ker \varphi), g'(ker \varphi) \in G/ker \varphi$, for some $g,g' \in G$, we have $$\phi [g(ker \varphi) g'(ker \varphi)]=\phi [(gg')(ker \varphi)]=\varphi(gg')=\varphi(g) \varphi(g')=\phi [g(ker \varphi)] \phi [g'(ker \varphi)],$$ so that it is a group homomorphism. Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$ Prove the identity:  $$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$  I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$  but got stuck along the way. Is there is any reason to make the permissions on an overridden C++ virtual function different from the base class?  Is there any danger in doing so?  For example:  class base {     public:         virtual int foo(double) = 0; }  class child : public base {     private:         virtual int foo(double); }   The  says that it is a bad idea, but doesn't say why.  I have seen this idiom in some code and I believe that the author was attempting to make the class final, based on an assumption that it is not possible to override a private member function.  However,  shows an example of overriding private functions.  Of course  recommends against doing so.  My concrete questions:   Is there any technical problem with using a different permission for virtual methods in derived classes vs base class? Is there any legitimate reason to do so?