I know that: $$\lim_{x \to 0}\frac{\sin x}{x} = 1$$  However all proofs I have seen resort to some kind of geometric appeal or otherwise fall to L'Hôpital's rule or Taylor series. However, I want an to see an proof that is purely algebraic (I mean least geometric intuition, mainly not diagrams to convince the reader of something, but obvious algebraic statements one after another) that uses nothing advanced like series and [I mean accessible to ordinary calculus beginner]... I do not want want elipson delta proofs but something that has quite evidently, no gaps.   Edit: Some comments have said sine cannot be defined without geometry or series. Well, I meant to say that we can assume that we know the unit circle definition. I only meant to say that we do not rely on diagrams in the proofs. We know the obvious facts that what sine  and radians are, etc. I did not mean it to take to formally and rigorously.    Hope you guys have some elegant and delightful proof of the kind. It is something which is bothering me for quite a while.
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.
Is there a way to search for a question (or an answer) that mentions a tag? How do I cd into a directory in the home folder? How do I change directories to downloads, from the home folder, in the latest version Ubuntu 12.10? I tried cd /downloads, but it gave an error saying that there is no such directory. What is $\mathbb{E}[W(s)\mathrm{e}^{W(S)}]$ where W(S) is a standard Brownian Motion? I know that since it is a Brownian motion that $W(s)\sim{N(0,s)}$. One approach I tried, was to see that $W(s)-W(0)\sim N(0,1)$ and to let $X = W(s)$ and then solve $\mathbb{E}[X\mathrm{e}^X]$ but I end up with some awful integral. I was wondering whether there was an easier way? Or perhaps a simple way to solve this integral: $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2{\pi}s}}x\mathbb{e}^{x}\mathbb{e}^{-\frac{x^2}{2s}}$ Marketing Cloud Mobile Push SDK - Switching app user I'm currently building a clienteling app that integrated the Mobile Push SDK on both iOS and Android and encountered some issue on switching the account.  I found that when I login to user A, and set contact key to A, the Inbox Messages will be send to this device. However, I logout user A and switch to a user B, I should set the contact key to B. But seems the inbox messages are associated with the Device instead of Contact, I can still see the inbox messages that were sent to contact A previously.  My question is, how should we deal with this situation, to prevent from seeing the previous inbox messages if the contact key has been changed? Why is the sky not purple? I realise the question of why this sky is blue is considered reasonably often here, one way or another.  You can take that knowledge as given.  What I'm wondering is, given that the spectrum of Rayleigh scattering goes like $\omega^4$, why is the sky not purple, rather than blue?    I think this is a reasonable question because  (or, strictly, violet or indigo) in rainbows, so why not across the whole sky if that's the strongest part of the spectrum?  There are two possible lines of argument I've  and I'm not sure which (if not both) is correct.  Firstly, the Sun's thermal emission peaks in the visible range, so we do actually receive less purple than blue. Secondly, the receptor's in our eye are balanced so that we are most sensitive to (roughly) .  Our eyes are simply less sensitive to the purple light than to the blue. Can you attack with different weapons using Extra Attack? Styling each feature of a layer individually I’ve a PostGIS table which is published as a Geoserver layer. Every feature in the table has some attributes that should style that feature (e.g color, size, opacity, external graphic url, ... etc).  I can create an SLD document and let it read its styling attributes from the table using &lt;PropertyName&gt;, but the problem is more complex than this, for example: some points features need to be styled as a simple circles, while some other need to be styled as external-graphic images. And from the SLD Reference, the SLD PointSymbolizer must have either &lt;Mark&gt; or &lt;ExternalGraphic&gt; element, for that reason I should create two rules one for the simple point symbolizer and the other for the external-graphic point symbolizer, and should add a filter to each rule so that it deals only with the appropriate features, something like:  &lt;FeatureTypeStyle&gt;     &lt;Rule&gt;         &lt;ogc:Filter&gt;             &lt;ogc:PropertyIsEqual&gt;                 &lt;ogc:PropertyName&gt;is-external-graphic&lt;/ogc:PropertyName&gt;                 &lt;ogc:Literal&gt;0&lt;/ogc:Literal&gt;             &lt;/ogc:PropertyIsEqual&gt;         &lt;/ogc:Filter&gt;         &lt;PointSymbolizer&gt;             &lt;Graphic&gt;                 &lt;Mark&gt;                     &lt;WellKnownName&gt;circle&lt;/WellKnownName&gt;                     &lt;Fill&gt;                         &lt;CssParameter name="fill"&gt;                             &lt;ogc:PropertyName&gt;fill-color&lt;/ogc:PropertyName&gt;                         &lt;/CssParameter&gt;                     &lt;/Fill&gt;                 &lt;/Mark&gt;                 &lt;Size&gt;6&lt;/Size&gt;             &lt;/Graphic&gt;         &lt;/PointSymbolizer&gt;     &lt;/Rule&gt;     &lt;Rule&gt;         &lt;ogc:Filter&gt;             &lt;ogc:PropertyIsEqual&gt;                 &lt;ogc:PropertyName&gt;is-external-graphic&lt;/ogc:PropertyName&gt;                 &lt;ogc:Literal&gt;1&lt;/ogc:Literal&gt;             &lt;/ogc:PropertyIsEqual&gt;         &lt;/ogc:Filter&gt;         &lt;PointSymbolizer&gt;             &lt;Graphic&gt;                 &lt;ExternalGraphic&gt;                     &lt;OnlineResource                xlink:type="simple"                xlink:href="smileyface.png" /&gt;                     &lt;Format&gt;image/png&lt;/Format&gt;                 &lt;/ExternalGraphic&gt;                 &lt;Size&gt;6&lt;/Size&gt;             &lt;/Graphic&gt;         &lt;/PointSymbolizer&gt;     &lt;/Rule&gt; &lt;/FeatureTypeStyle&gt;   My question is: Is there any better solution that is more dynamic? Where can I safely store my stuff? Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon? Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?  If so, can quantum mechanics describe the universe at high energies where there is no spacetime?  Schrodinger equation assumes the presence of time dimension. So how can such equations be valid at extremely high energy?  And is it meaningful to do physics if there is no time dimension?  Would the concepts of Hilbert space, wavefunction collapse etc be useful in that case? Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . I flagged a post and (choose one)   my flag was marked as helpful, yet nothing was done my flag was marked as declined, yet the post was deleted/edited/closed   Why? Regular cast vs. static_cast vs. dynamic_cast I've been writing C and C++ code for almost twenty years, but there's one aspect of these languages that I've never really understood. I've obviously used regular casts i.e.  MyClass *m = (MyClass *)ptr;   all over the place, but there seem to be two other types of casts, and I don't know the difference. What's the difference between the following lines of code?  MyClass *m = (MyClass *)ptr; MyClass *m = static_cast&lt;MyClass *&gt;(ptr); MyClass *m = dynamic_cast&lt;MyClass *&gt;(ptr); the cone is contractible I use Kali Linux and discovered the following "problem":  root@kali:/proc/self# ls -l cmdline -r--r--r-- l root root 0 May 27 13:43 cmdline root@kali:/proc/self# chmod 744 cmdline chmod: changing permissions of `cmdline': Operation not permitted   I am not allowed to change the permissions of that file... but I am root? Why is that? Are there ways to work around this? I allready tried some things like moving the file (what is obviously not possible), change the runlevel etc. How to watch videos in Amazon Prime Instant Video? How can I typeset this accented, dotless i? See the rough sketch above. I am able to typeset the accented i shown in the middle (for which I used the \v macro) and, obviously, the plain and simple i shown on the left also. However, I can't figure out how to typeset the one shown on the right. How can I do that? Find $f(x)$ for $f(f(x))=\sin x$ Find all (if that's possible) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x))=\sin x.$$  This is a problem a friend gave me. He said he saw it on a book a long time ago, but he couldn't remember what book. I literally have no idea how to solve this so I can't post my attempt here. Model won't follow its armature in pose mode I've set up a rig and built it around my model, and I've linked it to my model but when I rotate the bones in pose mode, the character doesn't move with them.   My rig is the human meta rig that comes with rigify. I've seen tons of videos of it working. Why isn't it working on mine?  My character was made by creating different body parts and joining them together into the same object, then a texture was added by unwrapping them in texture mode. Make IN clause behave like AND clause I've asked the same question on , but got no answer, so that's why I'm asking it here (I've read that simple sql questions should be asked at SO, but I have no choice).  Suppose I have 3 tables: posts, post_categories and categories. I'm implementing some kind of page with filters where I can filter posts by categories. User can select multiple categories. When he selects more than one, they should be summed. I can't get it working. What I have now is simple IN() SQL clause:  SELECT posts.id, post_categories.category_id FROM posts  JOIN post_categories ON posts.id = post_categories.post_id  WHERE post_categories.category_id IN (1,2,3) LIMIT 10;   But it is not matching all ids, it is OR and I need AND. What I need here is to find all posts that have categories with id=1 AND id=2 AND id=3, but instead this query returns posts that have at least one category from the IN() list. Installed Ubuntu 13.04 on a Dell laptop I had laying around.  Install goes well but when I'm done, I only get a command line, no GUI at all.  Tried several versions, tried 32 vs 64 bit all give me the same result.  Any ideas on what to do/try?