What is the limit of $\lim\limits_{x→∞}\frac{\sin x}{x}$
Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$
Why $SU(3)$ and not $U(3)$? Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead? Grep: unexpected results when searching for words in heading from man page I'm running into weird behavior when trying to grep a man page on macOS. For example, the Bash man page clearly has an occurrence of the string NAME:  $ man bash | head -5 | tail -1 NAME   And if I grep for name I do get results, but if I grep for NAME I don't:  $ man bash | grep 'NAME' $ man bash | grep NAME   I've tried other uppercase words that I know are in there, and searching for SHELL yields nothing whereas searching for BASH yields results.  What's going on here?  Update: Thanks for all the answers! I thought it worth adding the context in which I ran into this. I wanted to write a bash function to wrap man and in cases where I've tried to look up the man page for a shell builtin, jump to the relevant section of the Bash man page. There might be a better way, but here's what I've got currently:  man () {   case "$(type -t "$1")" in     builtin)       local pattern="^ *$1"        if bashdoc_match "$pattern \+[-[]"; then         command man bash | less --pattern="$pattern +[-[]"       elif bashdoc_match "$pattern\b"; then         command man bash | less --pattern="$pattern[[:&gt;:]]"       else         command man bash       fi       ;;     keyword)       command man bash | less --hilite-search --pattern='^SHELL GRAMMAR$'       ;;     *)       command man "$@"       ;;   esac }  bashdoc_match() {   command man bash | col -b | grep -l "$1" &gt; /dev/null } Why was C-3PO's memory wiped, but not R2-D2's? In Revenge of the Sith after Luke and Leia are born, they are sent to separate parts of the galaxy. The droids are given to Captain Antilles, with the instructions to clean them up, and have C-3PO's memory wiped. Why just C-3PO's and not also R2-D2's?   Initially I thought it was due to R2-D2's inability to communicate verbally, but he not only can communicate via text message (as seen in The Empire Strikes Back I believe), but we see other characters understanding his blips and bleeps as well. Upgrade SQL Server 2000 database to 2008 R2 and enable new features I have recently upgraded SQL Server 2000 database to 2008 R2.    What I did was:   Shutdown SQL Server 2000 (express) service on old machine, Move datafiles (mydatabase.mdf and mydatabase.ldf) to new machine, Run SQL Server Management Studio 2008, Connect to local database engine, Attach datafiles to database. Change the compatibility level of database to SQL 2008 (100).     Question: What else should I do to make migration complete?  I want to:   use new features like checksumming and full recovery model, make this database to be exactly as it was created in SQL 2008 R2, make this database to be fully compatibile, correct and be perfect suited for new, SQL 2008 R2 database engine.     In other words: I just want to know how to correctly and completly convert old SQL 2000 database to new 2008 R2 database, be calm that everything is done right and be happy with all new features.    I'm asking this question, because I've found a lot of sites on the Internet that says so many different things that makes me confused: some say that it is required to rebuild indexes, another says to do other things... and now I know nothing so I want to hear experienced person opinion and clear, step-by-step instructions. I work for very small company, I'm on my own and I don't want to screw things up.    Sir, I'm really impressed with your answer, I wasn't expecting so much.    So some comments:   The database is now in production. As I said, it was upgraded using deattach-attach method as I desribed in first post and as described on MSDN:  It had to be done quickly, so I was forced to do it that way. Let's forget about how inappropriate could it was and focus on current situation. The users/persmission is not a problem here - there are only few and permissions are simple. The application which use database is compatibile with SQL 2000 till 2012 so this is not a problem either. The database file (MDF) isn't big - only about 1GB.     Few more questions:   You recommend to use backup/restore method, but I did as written above, so can I encounter any problems now? Everything worked without any problem. About checksumming and full recovery model: it wasn't available/enabled on SQL 2000, so I want to use them now. You said that the only thing I need to do is to enable those options in database properties? I've read somewhere, that it isn't enough and I should also rebuild indexes or something. I really don't know, I just asking. I'm prepairing to migrate this database to SQL 2012 - so first it was from SQL 2000 to 2008 R2, now it will be from 2008 R2 to 2012 (it was impossible to do this directly because of lack of support of SQL 2000 databases in SQL 2012). So I understand that I should follow your guide: backup it in 2008 R2 and restore in 2012, then do the rest of your tips, right? Please explain me the backup/restore method: Is it like a dump of database to SQL queries and then restoring it by executing a bunch of queries? Will this method by the way "defragment" my database? If not, how to defragment/optimize it manually? As we were using SQL 2000 Express for years (no management interface), we were doing backups simply by stopping engine and RAR the DATA directory. For now, as we are on SQL 2008, isn't this still better than using backup function in Management Studio? Full recovery mode with frequent Transaction log backups - Where is the Transaction log stored - is it the LDF file? How sould I backup it properly?     I know that my questions may sound silly, I'm not professional database admin, but I am the only person here who can do such "hard core" task like upgrading the database engine. I'm also sure that your knowleadge will help a lot to other people like me.  Thank you very much for your time and knowleadge, I really appreciate this. I have a computer with Windows XP installed.  My user does not have a password.  When Windows starts, it goes to the login screen. Because my user has no password, is there any way to bypass this screen, and log in automatically? Problem with Refsection I can't seem to get the reference within my {refsection}.  \documentclass[a4paper]{article}  \newcommand{\myparagraph}[1]{\paragraph{#1}\mbox{}\\}  \usepackage[     backend=bibtex,     isbn=false,     url=false,     doi=false,     eprint=false, ]{biblatex}  %Point to the bibliography db \addbibresource{mybib.bib}  \begin{document}  \include{myparagraph}  \end{document}   My paragraph file  \myparagraph{\fullcite{dante1320}} \begin{refsection}  \cite{manzoni1827}  \printbibliography[heading=subbibliography] \end{refsection}   I get the first citation (dante1320) right, but the second prints as [manzoni1827]. I don't get any error when I run Bibtex concerning the missing reference. But I got it when I run Latex: LaTeX Warning: Citation 'manzoni1827' on page 2 undefined on input line 5. I use netstat to check my port status.  I was wondering what's the difference between port status LISTENING, TIME_WAIT, CLOSE_WAIT, FIN_WAIT1 and ESTABLISHED? Cannot suggest retag-only edit because "edits must be at least 6 characters" I was trying to suggest a retag of  on Seasoned Advice, but the system won't let me; I get the familiar message     Edits must be at least 6 characters; is there something else to improve in this post?   In general, that's not true for retagging; if you don't change the question body at all, it should work. I suspect the logic checking the edit is thrown off guard because of the title warning:     The question you're asking appears subjective and is likely to be closed       I found , but I cannot find any evidence (in SEDE) for invisible characters being present in the question's source. Also, that's supposed to be . Remove only the commas present within the double quotes In a text file, I want to remove , (commas) and also the " (quotes) (only if the double quotes  contains  numbers separated by commas).  56,72,"12,34,54",x,y,"foo,a,b,bar"   Expected ouput  56,72,123454,x,y,"foo,a,b,bar"   Note: I show the above line just as an example. My text file contains many lines like above and the numbers separated by commas present within the double quotes should vary. That is,  56,72,"12,34,54",x,y,"foo,a,b,bar" 56,92,"12,34",x,y,"foo,a,b,bar" 56,72,"12,34,54,78,76,54,67",x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar","12,34,54" 56,72,x,y,"foo,a,b,bar","12,34,54","45,57,84,92","bar,foo"   Expected output:  56,72,123454,x,y,"foo,a,b,bar" 56,92,1234,x,y,"foo,a,b,bar" 56,72,12345478765467,x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar",123454 56,72,x,y,"foo,a,b,bar",123454,45578492,"bar,foo"   There a n number of numbers present within the double quotes separated by commas. And also leave the double quotes which contains characters as it is.  I love sed text processing tool. I'm happy if you post any sed solution for this. It stands to reason: if it didn't, the entire sky would be covered with stars shining blindingly day and night.  But what causes a light wave (or an electromagnetic wave) to fade if there are no celestial bodies in its way? Or, alternatively, why doesn't it fade sooner? Some stars are visible to the naked eye that are hundreds of light years away. Others, closer, can only be detected with the aid of a telescope. Why? I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you tell $N\subseteq M_i$ for some $i$?  I was first trying to show it just in the case $N\subseteq M_1\cup M_2$, and hoping to extend it to finitely many $M_i$. If either of the $M_i$ contains the other, the claim follows, so I suppose neither $M_i$ contains the other.   In hopes of a contradiction, I suppose $N\not\subseteq M_1$ and $N\not\subseteq M_2$, Picking $x_1\in N\setminus M_1$ and $x_2\in N\setminus M_2$, I'd have $x_1+x_2\in N\subseteq M_1\cup M_2$. The only thing I could conclude was that actually $x_1,x_2\notin M_1$ and $x_1,x_2\notin M_2$, which seems like a dead end. What's the right way to do this? How do I get the Glare node to output transparent instead of Black background Insert results of a stored procedure into a temporary table In the Final Battle, when you remove doom tokens for rolling successes against an Ancient One do you stop counting successes for that roll? In Arkham Horror you have to roll a cumulative number of successes equal to the number of players to knock off a doom token from the Elder God's health. If you reach the number of players worth of successes do you then stop counting for the player and move on or do you count the extra successes against the next doom token?  We played it so that you kept on adding them but it meant that we turned Nyalothotep into mince meat quite quickly (One Character was blessed had a fight of 6 a shotgun and Fight +1 skill card though) Render not showing UV texture I'm glad the outside overlay scrollbars are finally gone.  But the scrollbar is still a razor thin line that becomes a less razor thin scrollbar, but still thin scrollbar, when the mouse hovers.  How can I get a normal scrollbar? I.e., like in other OSes, like in Firefox, a scrollbar that doesn't change depending on what the mouse points to, and with a nice width.  I already tried  gsettings set com.canonical.desktop.interface scrollbar-mode normal   but no luck. How to subtract a day from a date? I have a Python  object. What is the best way to subtract one day? public class doublePrecision {     public static void main(String[] args) {          double total = 0;         total += 5.6;         total += 5.8;         System.out.println(total);     } }   The above code prints:  11.399999999999   How would I get this to just print (or be able to use it as) 11.4? Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$? Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any such matrix $A$, the set $K[A]$ of polynomials in $A$ is a commutative subalgebra of $M_n(K)$; the question is whether for any pair of commuting matrices $X,Y$ at least one such commutative subalgebra can be found that contains both $X$ and $Y$.  I was asking myself this in connection with frequently recurring requests to completely characterise commuting pairs of matrices, like . While providing a useful characterisation seems impossible, a positive anwer to the current question would at least provide some answer.  Note that in many rather likely situations one can in fact take $A$ to be one of the matrices $X,Y$, for instance when one of the matrices , or more generally if its  (so coincides with the characteristic polynomial). However this is not always possible, as can be easily seen for instance for diagonal matrices $X=\operatorname{diag}(0,0,1)$ and $Y=\operatorname{diag}(0,1,1)$. However in that case both will be polynomials in $A=\operatorname{diag}(x,y,z)$ for any distinct values $x,y,z$ (then $K[A]$ consists of all diagonal matrices); although in the example in  the matrices are not both diagonalisable, an appropriate $A$ can be found there as well.   I thought for some time that any maximal commutative subalgebra of $M_n(K)$ was of the form $K[A]$ (which would imply a positive answer) for some $A$ with minimal polynomial of degree$~n$, and that a positive answer to my question was in fact instrumental in proving this. However I was wrong on both counts: there exist (for $n\geq 4$) commutative subalgebras of dimension${}&gt;n$ (whereas $\dim_KK[A]\leq n$ for all $A\in M_n(K)$) as shown in , and I was forced to correct  in the light of this; however it seems (at least in the cases I looked at) that many (all?) pairs of matrices $X,Y$ in such a subalgebra still admit a matrix $A$ (which in general is not in the subalgebra) such that $X,Y\in K[A]$. This indicates that a positive answer to my question would not contradict the existence of such large commutative subalgebras: it would just mean that to obtain a maximal dimensional subalgebra containing $X,Y$ one should in general avoid throwing in an $A$ with $X,Y\in K[A]$. I do think these large subalgebras easily show that my question but for three commuting matrices has a negative answer.  Finally I note that  to the cited MO question mentions a result by Gerstenhaber that the dimension of the subalgebra generated two commuting matrices in $M_n(K)$ cannot exceed$~n$. This unfortunately does not settle my question (if $X,Y$ would generate a subalgebra of dimension${}&gt;n$, it would have proved a negative answer); it just might be that the mentioned result is true because of the existence of $A$ (I don't have access to a proof right now, but given the formulation it seems unlikely that it was done this way).  OK, I've tried to build up the suspense. Honesty demands that I say that I do know the answer to my question, since a colleague of mine provided a convincing one. I will however not give this answer right away, but post it once there has been some time for gathering answers here; who knows somebody will prove a different answer than the one I have (heaven forbid), or at least give the same answer with a different justification. Do we know of any living forms on Earth that could continue to live on Mars?