What is   $$\lim_{x \to 0^+} f(x)\quad\text{ where }\quad f(x) =\left( \frac{\sin x}{x}\right)^{1/x^2}$$  ? The problem suggests to use a Taylor expansion, but it is not clear to me what to expand.
I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ?
I have a machine running Ubuntu which I SSH to from my Fedora 14 machine. I want to forward X from the Ubuntu machine back to Fedora so I can run graphical programs remotely. Both machines are on a LAN.  I know that the -X option enables X11 forwarding in SSH, but I feel like I am missing some of the steps.  What are the required steps to forward X from a Ubuntu machine to Fedora over SSH? I compile most of my documents via Makefiles which take care of the bibliography, indexes, etc. Often I'd like to compile different versions from the same TeX file changing only small things (e.g. a beamer presentation plus a version with the handout option).  Is there any standard mechanism for passing parameters to the TeX file so that I can just type make handout to get the handout version in the above example? A zero-dimensional ring is Noetherian? Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian.   Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$, contradiction. Hence every prime ideal is maximal. Consequently, every increasing (or decreasing) chain of prime ideals will be trivial in the sense that it will have length $0$. Hence the ring is Noetherian (and Artinian).  Is there a problem with the above proof? It'd be nice to see reputation updates in the app, as we get on both the desktop and mobile sites.  Mock-up of how this would look:        Other possible icon states:        While I understand reputation is shown in the feed, it's a bit unwieldy, and has no notification.   With this feature implemented, the feed could arguably lose both reputation and inbox items, and be left with the interesting questions that fill it up. This would make for a much more consistent UX across app and desktop. Elliptic Coordinates - Inverting the transformation The standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$:  $x = a \cosh(\mu) \cos(\nu)$  $y = a \sinh(\mu) \sin(\nu)$  Is there any way to get the transformation $(x,y)$ to $(\mu,\nu)$? Meaning is there a way to find:  $\mu = f(x,y)$  $\nu = g(x,y)$  I'm guessing that it would involve $\sinh^{-1}$'s and $\cosh^{-1}$'s, if it was possible to do this at all. just after the Big Bang and before inflation the currently observable universe was the size of a coin. One millionth of a second later the universe was the size of the Solar System, which is an expansion much much faster than speed of light. Can space expand with unlimited speed? I need help with a specific command that I am using in a new game that I am creating inside of Minecraft. The command is:  /give @p sign 1 0 {BlockEntityTag:{Text1:"{"text":" [MF2]","color":"gold","clickEvent": {"action":"run_command","value":"tellraw @a [""{"text":"[MF2]  ","color":"gold"},{"selector":"@p","color":"dark_aqua"},{"text":" is now an admin of the server!","color":"white"}]"}}",Text3:"{"text":"Click to join Admin","color":"dark_aqua","clickEvent": {"action":"run_command","value":"scoreboard teams join admin @p"}}"},display: {Name:"Custom Sign"}}   If I take out the /tellraw command then the command works fine, but I need the command in there to announce to the game that a new admin has joined.  With the tellraw command, the error message is the following:  Data tag parsing failed: Unexpected token 't' at: text":" [MF2]","color":"gold","clickEvent":{"action":"run_command","value":"tellraw @a [""{"text":"[MF2] ","color":"gold"}, {"selector":"@p","color":"dark_aqua"},{"text":" is now an admin  of the server!","color":"white"}]"}}",Text3:"{"text":"Click to join Admin","color":"dark_aqua","clickEvent": {"action":"run_command","value":"scoreboard teams join admin @p"}}"   If anyone knows how to fix this, please let me know. How to effectively close a page? Before asking my question, I'll use an example to clarify the idea.  Say that a company used to sell horses and then they turn into a sheeps business. While some of the old pages can be mapped to the new pages using a 301 redirect, because they're really related in the content and the company don't want to lose users and link juice, there are a lot of pages strictly related to horses that they want to shutdown.  What's the best practice? Is there a way to say to users and Search Engines the content you were looking for won't be available any more?  Basically a 301 redirect of the closed pages to new ones seems not so good, because you'll show a content totally unrelated to the searched one.  P.S: I'm aware that probably the best fallback is to simply map those pages to the homepage but I hope in a smarter solution. "This product is good value" vs "This product is of good value" in a sentence 1.This product is good value. 2.This product is of good value.    Are there any difference in meaning between "good value" and "of good value"? How to Map a subset of list elements to a function? Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree.  HINTS ONLY! Say, I'm given a filename like libasound.so.2. Is there a way to query the package manager to correlate the filename with its corresponding package?  I'm interested to get answers for any package manager, but my primary focus is on apt and yum. Get specific GPS location somehow on windows 8 File permissions and saving Pre-generate Review audits to avoid the lag? How to reset main user account? My main account got messed up, as I tried to fix it things went downhills.   keyboard mess - tried unity --restore and deleting .gconf2*, fail deleted .* in ~, total havoc. Chromium and Firefox crashes etc.   Now, another user account is fully functioning, but I really want my regular username back.  Is there any way to completely wipe the settings for the messed up account or copy the profile from the functioning user? In Visual Studio, How do I show all classes inherited from a base class?    For example, in ASP.NET MVC there are several '' types -- and they all inherit from / implement the base class ActionResult.    It looks like unless you just 'know' that View and Json are valid ActionResult types, there is no way you can easily find this information out.    Please prove me wrong.  Is there something in the object browser that makes this easy to find out?  I'm even up for suggestions of tools outside of Visual Studio to discover this information about various classes.  For example: is there something in Resharper that will help me out? How can I count all the lines of code in a directory recursively? Why is the gate charge curve (Miller plateau) of MOSFETs dependent on Vds? SELinux preventing Apache from writing to a file SELinux is preventing the apache user from writing to a log file which it owns. When I do setenforce 0 it works. Otherwise it shows this error  IOError: [Errno 13] Permission denied: '/var/www/webapp/k/site/k.log'   The security context of the file:  $ ll -Z k.log  -rw-r--r--. apache apache system_u:object_r:httpd_sys_content_t:s0 k.log   The file was created when the SELinux mode was set to permissive.  How to set the security context so the apache user can write in that directory? I did set that directory security context using chcon but I can't find a suitable file type.  From audit.log:  type=AVC msg=audit(1409945481.163:1561): avc:  denied  { append } for  pid=16862 comm="httpd" name="k.log" dev="dm-1" ino=201614333 scontext=system_u:system_r:httpd_t:s0 tcontext=system_u:object_r:httpd_sys_content_t:s0 tclass=file type=SYSCALL msg=audit(1409945481.163:1561): arch=c000003e syscall=2 success=no exit=-13 a0=7fa8080847a0 a1=441 a2=1b6 a3=3 items=0 ppid=15256 pid=16862 auid=4294967295 uid=48 gid=48 euid=48 suid=48 fsuid=48 egid=48 sgid=48 fsgid=48 tty=(none) ses=4294967295 comm="httpd" exe="/usr/sbin/httpd" subj=system_u:system_r:httpd_t:s0 key=(null)