I need a rigorous proof that verify why the limit of $\dfrac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$.  I tried before but i do not know how start this proof. I would appreciate if somebody help me. Thanks.
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.
If I simply wrap my query with:  BEGIN TRANSACTION    COMMIT TRANSACTION   If anything fails inside of that, will it automatically rollback?  From looking at other code, they seem to check for an error, if there is an error then they do a GOTO statement which then calls ROLLBACK TRANSACTION  But that seems like allot of work, to have to check for IF( @@ERROR &lt;> 0) after every insert/update. Let $X \sim Bin(n,p)$ show that $X/n \nsim Bin()$ Let $X \sim Bin(n,p)$   Part A.  Show that the argument      "Then $X/n \sim Bin()$ with $E[X/n]=p$and $Var[X/n]=pq/n$"    is false.   My book says we can prove this result using a moment generating function. But to me it makes no intuitive sense: We have Bin-distributed RV $X$ and if we rescale it with $1/n$ of course it will be Bin-distributed.  And we know from standard formulas that $E[aX]=aE[X]$ and $Var[aX] = a^2Var[X]$ so I think the argument is correct.   (For full disclosure, the argument we should falsify according to the book is "Let X be a binomial random variable with mean np and variance np(1−p). Show that the ratio X/n also has a binomial distribution with mean p and variance p(1−p)/n." But I think the argument is correct. )    Ok, now I get part A. Here is Part B. Prove it using the moment generating function of $X/n$.   Attempt: I get $$m_X(t)=E[\exp(tX/n)]=(m_X(t))^{1/n}$$ How do I deal with a bored or distracted player bringing down the group? My group have been playing various D&amp;D style games for a few years,  I have one player however that often will drift off and start playing on his phone  and several times has actually fallen asleep (with his character having to be "nudged" along by his fellow players).  He insists he's "just tired" and "isn't bored" — and besides, he keeps insisting we all get together for D&amp;D and wants to play.  Unfortunately we're all pretty busy, so if I'm to get the group together at all it's usually late on a Sunday. After talking to the player he insists it's not the game, or the story, he just happens to be tired. But, it's the same excuse every week, and the odd times we've played at different times or situations, the falling asleep has happened.  How do I cope with:   The morale destroying (for me) idea of your players falling asleep through a campaign and story you've put lots of work into. The general tone of the room when this player loses interest. (Occasionally a couple of others zone out into Facebook when they're not "active" because the players have split up / gone unconscious).   Given that it's an "all or nothing" situation with the group (there are only 3 players and they won't play unless all 3 are around), how do I best deal with it? I know that the Naruto movies are outside the timeline in the sense that they don't affect whatever happens after them. However, they must fall somewhere in the Naruto timeline — based on who are in the movies and how the characters look like, etc. For example, if Sasuke isn't in the movie, it should take place after he left. For each of the movies, approximately what timeframe (between episodes/arcs) are they set?  The movies:   Ninja Clash in the Land of Snow Legend of the Stone of Gelel Guardians of the Crescent Moon Kingdom Naruto Shippuden the Movie Bonds The Will of Fire The Lost Tower Blood Prison Road to Ninja The Last - Naruto the Movie Boruto: Naruto The Movie How can I change the default date format? When I paste a date into Excel 2010 in the format "2012-12-07 00:00:00.000" I'd like it to be displayed by default as "2012-12-07", but it is actually displayed as "00:00.0".   I know I can work around this by prepending a ' or changing the cell format, but I'm interested in changing the default format.  An approach that almost works is going to Control Panel->Region and Language->Long Time format and changing it to something like "h". This results in Excel displaying the dates I paste in as "2012-12-07 00:00:00.000", but screws up times displayed outside of Excel.  The dates I'm pasting in are from SQL Server. Why does the set of all singleton sets not exist? Proof conclusion to: Let m,n $\in$ $\mathbb{N}$ $\backslash$ $\{0\}$ . Prove that $s |\tau (m^{s}) - \tau(n^{s})$, where $\tau$ is a divisor function.   Hello, so I have a tiny question about the summarization of my proof.  I got to the point where I defined my $\tau$ divisor function and got them both $\tau(m^s)$= $\prod$ something +1 and $\tau(n^{s})$ = $\prod$ something +1.    I've noticed that the last terms of the polynomial are both one. Why is it with the +1 part, I can conclude with that:  s| $\tau(m^s)$ - $\tau(n^s)$.   The line for why that is, isn't quite clear. Hyperref warning - Token not allowed in a PDF string Bound for the correlation of three random variables There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? Is it legal to copy Stack Overflow questions and answers? I just found http://programmingfaq.w3ec.com/faq/4761/whats-the-hi-lo-algorithm which has an exact copy of this Stack Overflow question, , with all its answers and no difference in a single character. There is no reference to Stack Overflow.  Is this legal, or at least tolerated? Is this impertinent theft of information? I just can't see any sense in copying whole pages, particularly when not referencing the source. With the advent of the WordPress Answers Stack Exchange site, there's undoubtedly going to be overlap.  But I just want to get some advice:  I'm a developer and I kinda hate WordPress, but have to use it occasionally. My WordPress questions are usually development ones (when I have them), and most often related to 'how WordPress handles things', which are pretty obvious in straight programming terms.  But with the WordPress layer of abstraction, they become problematic.  I find that most developers that have gone near WordPress are usually also PHP/JS/CSS/MYSQL guys who dabble with WordPress and, as a result, hangout more on SO. And theres a die hard bunch who are heavy into the WP.SE site, but generally the WP site is for people trying to muddle through the day to day operation of using WordPress and its many themes/plugin etc.    As a result, if I were to ask something pretty in-depth about writing / integrating new &amp; strange features or something, I'm talking to a much smaller group of devs than I would be on SO.  So, enough of the ramble, my question:  Should I be posting heavy development questions relating to WordPress on the WP.SE site or to SO? Can I post to both, or is cross posting frowned upon in these cases still? I've recently updated to Ubuntu 15.04 and after installing the latest nvidia drivers (346.59) which worked on ubuntu 14.10 seems to have caused some trouble on boot.  All I get is the following when booting:   A purple screen for about a second and then  [0.514409] ACPI PPC Probe failed. Starting version 219  A black screen with a message of my monitor saying : No Signal detected.   I am using a GTX 970. I'm trying modeling things like this using oracle:  An account can have 1~5 members  So I'll have a account table and a member table with a FK of account. But is that possible to check if the account have less than 5 members?  I've tried constraint and trigger but oracle doesn't allow sub queries in constraint or condition of trigger like  CHECK( SELECT ...) or IF count(*) &gt;= 5  which make things getting hard. Named Credentials - Merge Fields not Resolving How do I call one constructor from another in Java? Is it required for panel data to use dummy variables? I am doing research considering seven countries and I have panel data. My question is: do I need to include dummy variables every time I use panel data in regression, or is enough to do it as a fixed effect in Stata, without dummy variables? Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $ Prove that if $a&gt;0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $.   Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By factoring $(a^{10}-1)(a-1) \geq 0 $.  I am not sure how to proceed any further. Thread Confinement Sending `time` command into text file I want to get the result of time command into a text file but it's not working it only put blank space in the text file.  I already tried this commands,  A-  $ x=`time` $ echo $x &gt; log.txt $ cat log.txt  $   B-   $ time &gt; log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s $ cat log.txt  $   C-  $ time &gt; log.txt 2&gt;&amp;1  real    0m0.000s user    0m0.000s sys     0m0.000s  $ cat log.txt  $    What I really want is this.   $ time &gt; log.txt $ cat log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s I'm writing a mildly dystopian scenario where competition is taken to vicious extremes, and I was wondering if there was a word that I could use (specifically a noun that was a synonym of "competition") to describe the sort of vicious, cutthroat ... _________ (the word that I want) that was going on.  I'm specifically looking for a word that doesn't refer to a one-off event (like "contest" or "tug-of-war"), but rather is more about the state of things.