A student I tutor asked me this question. The L'Hopital's approach is quick and easy, but is there a direct but rigorous way to show this without invoking L'Hopital's?
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.
Any way to see mail sent by other user through shared address? My wife and I have a "shared" email address, like thesmiths@example.com which we both use from Gmail when we both want to get replies or for signing up for certain websites.  When an email is received at that address, it gets forwarded to both of our personal Gmail accounts.  The problem is that when either one of us "sends as" that address, the other does not see the sent message. So we have to cc or bcc each other (and she usually forgets.)  Is there any automagic way to have those sent emails appear in BOTH of our Gmail accounts?  I've tried filters but I don't think they work on outgoing messages. Basically we want a filter that would automatically forward emails from that address to the other person.  I know this can be done with standalone mail clients, but neither of us uses those. I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.  Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false? Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization? How did this person gain possession of the Infinity Stones? Why aren't the humans wearing environmental suits in Star Wars spacecraft? Significance of starting the Fibonacci sequence with 0, 1.... DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.  QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?  For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence?  Is there a general mathematical explanation for the relationship between any sequence represented by $a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2]$ and the Fibonacci sequence?  Or, back to my example sequence, is there a general mathematical relationship between:  $2,7,9,16,25,41,66,107,173,280...$  and  $0,1,1,2,3,5,8,13,21,34...$  Perhaps the Golden Ratio explains it somehow?  Any help would be appreciated. I'm having a problem setting up a vhost on Apache 2.4, using CentOS 6.3.  This question seems to have been answered several times on this forum, but none of the solutions work for me.  Basically, I set up a vhost with a particular document root. When I attempt to access the site, I get a permissions error. But I have no problem if I use a different root.  More specifically, if I set the root to be /volume1/web/xxx, I get a 503 because of a permission problem on index.html.  But if I set it to be /volume1/web, the index.html comes up without a problem.  The details:   I set the persmissions on volume1, web, and xxx to 755. I set the permissions on the index.html files to 777. I copy the index.html file to both /volume1/web and /volume1/web/xxx, so that they have identical index files with identical permissions.   In httpd.conf, I set the VirtualHost options so that I can easily switch the document root by changing 2 comment marks  I have the following lines in the conf file:  NameVirtualHost *:80  &lt;VirtualHost *:80&gt;     ServerName localhost     #DocumentRoot /volume1/web     DocumentRoot /volume1/web/xxx      #&lt;Directory /volume1/web&gt;     &lt;Directory /volume1/web/xxx&gt;       Options Indexes FollowSymLinks Includes ExecCGI       AllowOverride All       Order Allow,Deny       Allow from all     &lt;/Directory&gt;      ErrorLog logs/localhost-error_log     CustomLog logs/localhost-access_log common &lt;/VirtualHost&gt;   These lines contain all the suggestions that worked for other people with the same problem.  What I get when I enter localhost in my browser depends on the document root.   If the root is /volume1/web, the index.html page comes up. if the root is /volume1/web/xxx, then I get a 503 error, and the apache error log has this line:  [Thu Jul 19 17:17:52 2012] [error] [client ::1] (13)Permission denied: access to /index.html denied   I also tried adding an index.php file to the directory. I then got permission denied messages for both the php file and the html file.  I also tried setting the user : group to apache : apache on the directories and files. That didn't help.  I've done everything I can think of to fix the permissions, to no avail.  Does anyone have an idea about what's happening here?  Thanks. Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$ Catching poisoned arrows When catching an arrow tipped with poison (like the ones used by yuan-ti) with an ability like the one the monk has or the one given by arrow catching gloves: does the ability reduce the poison damage? If so, is the total damage or the piercing one caused by the arrow only the damage that must be downed to 0 to catch the missile? Macro with lstlisting inside I have been following  about using \NewDocuentCommand with lstlisting and the accepted answer is nice except it is too big and it doesn't quite work for me.   I came up with this example to test xparse:    \documentclass{article} \usepackage{xparse} \usepackage{listings}  \ExplSyntaxOn \NewDocumentCommand{\code}{m +v}{     \begin{lstlisting}[style=#1,numbers=none,xleftmargin=0.35cm,framesep=0mm]         #2     \end{lstlisting} } \ExplSyntaxOff  \begin{document}  \code{C}{ int x = 200; unsigned int y = 0; }  \end{document}   Like stated in the  I first used m as a normal parameter and then +v as a multiline verbatim parameter, but it fails. It looks like I am still missing something... Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$ Battery Ampere-hour rating vs Battery Amps (not an experienced user) I have almost no experience whatsoever with the technicals of electronics so this is probably a very easy question for someone who does.  Do you know if a battery with a 200Ah rating can put out 200A for one hour or are there limitations? According to this website's third paragraph (), you can.     For example, an average automotive battery might have a capacity of about 70 amp-hours, specified at a current of 3.5 amps. This means that the amount of time this battery could continuously supply a current of 3.5 amps to a load would be 20 hours (70 amp-hours / 3.5 amps). But let’s suppose that a lower-resistance load were connected to that battery, drawing 70 amps continuously. Our amp-hour equation tells us that the battery should hold out for exactly 1 hour (70 amp-hours / 70 amps), but this might not be true in real life. With higher currents, the battery will dissipate more heat across its internal resistance, which has the effect of altering the chemical reactions taking place within. Chances are, the battery would fully discharge some time before the calculated time of 1 hour under this greater load.   But wouldn't that mean you could hook up a 200Ah battery and ask it to put out 12000A for one minute or 720000A for a second?  That seems very unrealistic, lol.  I'm trying to find the proper kind of off grid battery that can power a microwave through a 3000 watt power inverter.  The microwave needs 1800 watts and the battery needs to be 12 volts so that would mean I need about 150 amps, I'm wondering if a battery with a 200Ah rating could do it? Intuitive explanation of convergence in distribution and convergence in probability What is the intuitive difference between a random variable converging in probability versus a random variable converging in distribution?  I've read numerous definitions and mathematical equations, but that does not really help. (Please keep in mind, I am undergraduate student studying econometrics.)  How can a random variable converge to a single number, but also converge to a distribution? How do I get a list of all files (and directories) in a given directory in Python? As , the retag privilege serves a lot more confusion and a lot less of its benefit due to the suggested edit system. It must be done away with. It will be done away with. ♪  In light of this decided course of action, we're going to need a new snack for the road between "Create Tags" at 300 reputation and "Established User" at 1000 reputation. We'd like your help in thinking about some ideas on what would be a nice privilege to sit at this level. Since we're... a bit dry on our end.  The primary guideline in deciding what kind of privilege to put here is acknowledging its position in the overall hierarchy of privileges. 500 reputation is an early-intermediate user, one who has started to really get the hang of the site and its function. It can be something that provides convenience in how they currently can act on the site, or it could be some new functionality that fits as an early ability (especially considering there's no new abilities until 2000 reputation's "Edit Everywhat" ability).  I look forward to hearing what creative ideas y'all can provide. I'm a C++ newbie, but I wasn't able to find the answer to this (most likely trivial) question online.  I am having some trouble compiling some code where two classes include each other.  To begin, should my #include statements go inside or outside of my macros?  In practice, this hasn't seemed to matter.  However, in this particular case, I am having trouble.  Putting the #include statements outside of the macros causes the compiler to recurse and gives me "#include nested too deeply" errors.  This seems to makes sense to me since neither class has been fully defined before #include has been invoked.  However, strangely, when I try to put them inside, I am unable to declare a type of one of the classes, for it is not recognized.  Here is, in essence, what I'm trying to compile:  A.h  #ifndef A_H_ #define A_H_  #include "B.h"  class A {     private:         B b;      public:         A() : b(*this) {} };  #endif /*A_H_*/   B.h  #ifndef B_H_ #define B_H_  #include "A.h"  class B {     private:             A&amp; a;      public:         B(A&amp; a) : a(a) {}  };  #endif /*B_H_*/   main.cpp  #include "A.h"  int main() {     A a; }   If it makes a difference, I am using g++ 4.3.2.  And just to be clear, in general, where should #include statements go?  I have always seen them go outside of the macros, but the scenario I described clearly seems to break this principle.  Thanks to any helpers in advance!  Please allow me to clarify my intent if I have made any silly mistakes! Let $G = (\mathbb{C} - \{0\}, \cdot)$, a subgroup $U = \{x+yi \mid x^2 + y^2 = 1\}$. Use the Fundamental Theorem to show that $G/(\mathbb{R}_{&gt; 0},\cdot)$ is isomorphic to $U$ where $\mathbb{R}_{&gt; 0}$ denotes the positive reals.  I am struggling to find the correct mapping from $G$ to $U$ that results in the the kernel being the nonnegative reals under multiplication. I'm hoping to get help with this mapping then I can show it is a homomorphism and onto and can use the Fundamental Theorem. How to run grep with multiple AND patterns? Error creating GeoPackage raster store in GeoServer Explanation of seasons in "A Song of Ice and Fire"