Find limit of $\lim_{x\to \infty} \sqrt{x^2+x}-\sqrt{x^2-x}$
Find the limit: $\displaystyle\lim_{x\to \infty} (\sqrt{x^2+x}-\sqrt{x^2-x})$.
How can I prove that if $n$ is even then $n^2$ is even and if $n$ is odd then $n^2$ is odd? How do I change my username? Some time ago, when I installed Ubuntu, I chose a rather stupid username for my account that I do not want to use anymore. How do I change this (including the name of my home directory, and the name in the terminal) without losing settings for applications? How do I keep permissions and my keys for various authentification (e.g. email, SSH, GPG and more)? What settings could possibly get lost if I changed my username? How do we control web page caching, across all browsers? Our investigations have shown us that not all browsers respect the HTTP cache directives in a uniform manner.  For security reasons we do not want certain pages in our application to be cached, ever, by the web browser. This must work for at least the following browsers:   Internet Explorer 6+ Firefox 1.5+ Safari 3+ Opera 9+ Chrome   Our requirement came from a security test. After logging out from our website you could press the back button and view cached pages. Travel on a Schengen visa stating EEA family member without spouse? Selecting one every n frames from a video using ffmpeg Green is the most common circuit-board color because it has become an industry standard.  What I'm interested in is How the traditional "PCB Green" become a standard in the first place?   Were there any interesting historical reasons for the initial choice, or was it just a product of what one particularly successful company was doing that became the de-facto standard? 2 different bibliographies I'm writing my thesis and I would like to make 2 "bibliography" references: The first one would be for the publications I'm in and the second one for external references. I've made one *.bib file right now and it's working but I have no idea how I should work with 2 *.bib files.  Any suggestion will be appreciated. Dealing with "Find out who's going to buy the croissants" On Stack Overflow at the moment, there is a popular question which is currently high in the Hot Questions feed:   So far it has been closed twice (once by the community and a moderator and once by a moderator alone) but re-opened twice by the community. It is clearly not a [so] Stack Overflow question. Reading between the lines, there could be a valid Software Engineering question there, but it would be a very basic one (at the fizz-buzz competency level) and looks more like a populist attempt to farm reputation than a serious question.  Worst of all, even though there are some interesting answers on this question, they are all going to be completely useless to future visitors to the site, since they will be completely undiscoverable. If the question were at least "What is the best fair distribution algorithm?" it would still be off-topic (primarily opinion based), but at least it might help someone once the topic falls off the Hot Questions feed.  Although this has turned into just a bit of fun, and I don't want to kill that, enough is enough. While this question is open, unlocked, and still seen as a valid question, it is setting a bad example for the whole Stack Exchange network.   questions like this, especially on the grandfather Stack Exchange site, devalue the work we are putting in on every other site by making it appear acceptable to re-open populist questions like this after two moderators have classified it as off-topic.  What should we do about this question?   Should it be left open? Power to the people? Closed and locked by a moderator? Migrated to another site? &hellip;or something else? I read this book series in middle school. I believe that it would be classified as young adult fiction/fantasy. A girl has magical powers, and with her growing powers comes runes/tattoos in her arms. This is shown on every book cover. I know that one cover is blue. This one shows the girl in a black tank top and black pants. One is a yellow cover, with her in a dress (I think).  The girl has a friend/mentor, who she develops feelings for. The girl is the key to stopping a war between multiple factions/people. Each side is trying to use her to their advantage. I think the series was written in the early 2000s. I am pretty much sure that for native speakers the issue I am going to bring up might look as an uncalled question as they can easily figure out which form of a verbal part of speech should be used, be it a gerund, an infinitive, or a bare infinitive. However, it can be pretty much misleading for a foreigner because the same construction works differently with different words and I don’t see a logical explanation. For example:  The purpose of this article is to analyze the issue…  My hobby is reading. One may assume, and many foreigners really come to this conclusion, that it’s OK to switch the “gerund” and “infinitive” as the meaning will stay the same, that is:  The purpose of this article is analyzing the issue… (same as 1 above)  My hobby is to read. (same as 2 above)   However, as I was told, this assumption is erroneous. Another example: His task was watching after them. (not good) His task was to watch after them. Is there any principle which governs the choice of the right form in such cases? Recently I answered two questions on SO: The  is a completely trivial answer to a really basic question (kind of "read the manual and follow the one liner example there"). On the other hand, my  is a much more complicated answer that requires much more specialized knowledge of a very specific domain (happens to be related to my PhD research).  Clearly, the first answer could have been given by many SO users - I just happened to be the first to pick up the glove on this question. However, this is not the case for the second answer.  In contrast, when it comes to reputation - the first answer instantly got upvoted (everyone can relate to a simple question/answer), while the second one - I don't believe it will get much upvotes...  So, life isn't fair: for the one answer that really very few people could provide (the second answer) and is much harder to find and get -- is not rewarded, while the trivial answer is rewarded fast and strong...  How can we amend this? How can we encourage rare and specialized answers for difficult and deep questions?  Do we want SO to turn into a pool of trivial and superficial questions?    Update: Thanks to many meta-users who took the time to read through this question and ponder its implication, both answers I used as examples got exposed far beyond their fair-share. So, despite the fact that my original claim that the specialized answer get less reps is no longer valid, this does not change fact that in general my claim is stil valid: specialized answers requiring expert knowledge will recieve less attention at first.    Bottom line: (following my wish to have some sort of ). Indeed it sometimes seems like life isn't fair, but this is only in the short term. In the long run high quality answers do get their proper credit and exposure and their fair share of reputation. The reputation system of SO works quite well balancing and accounting for many factors and parameters. It would require a really strong argument backed up with serious research to change this mechanism.   Thank you all for the lively discussion and for showing me that life is fair after all. I dragged a file to the recycle bin a few days ago. I have Windows 7 premium on a 245 GB partition, a 240 GB SSD, and a 250 SSD. And somewhere in the past few days I incidentally installed Delta Search. I know I should remove it right away, but I just switched to Windows on the partition of a 3.5 1 TB drive and would like to restore that file and deal with the malware after Christmas is over.  I'm pretty sure I know what drive that deleted files should be on, but what is the real name of the recycle directory and exactly where is it located? In other words, if I know the name of the drive what would the rest of the path look like so I can just paste it in my Explorer window and change the drive letter. Should 'drive by' downvoting be more effectively caught? I have an answer that I'm certain is technically correct. However, it continues to be downvoted. The answer has already been noted in comments on other answers to be better. It goes more in-depth than the others, yet it continues to be downvoted.  The already upvoted answers were nowhere close initially, and I had already commented on them to say why it is wrong...  Isn't there a limit for that? Shouldn't something be catching this? Why can't a single photon produce an electron-positron pair? In reading through old course material, I found the assignment (my translation):     Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.   My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.  How to solve this task? What is inertia of a body? The definition of  is  "Inertia is the resistance offered by the body whenever its state of motion is changed."   What is inertia of a body? Is inertia actually a force exerted by the body?  If so, then why is there no application of inertia in numerical problems involving application of force on a body? What is the actual application of inertia in theory? Arch on WSL: libQt5Core.so.5 not found, despite being installed I was hit by some hardcore power-tripping forum mods on the Arch Linux forum, so here I am.  I'm trying to run GUI apps on WSL, such as dolphin or kate. However, none of them can find libQt5Core.so.5, despite it being installed in /usr/lib as you'd expect. Here's a terminal transcript:  /u/lib » DISPLAY=:0.0 dolphin dolphin: error while loading shared libraries: libQt5Core.so.5: cannot open shared object file: No such file or directory /u/lib » pwd /usr/lib /u/lib » ls -la | grep Qt5Core -rw-r--r-- 1 root root     1143 Aug  7 06:05 libQt5Core.prl lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5 -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5.11 -&gt; libQt5Core.so.5.11.1* -rwxr-xr-x 1 root root  5234312 Aug  7 06:05 libQt5Core.so.5.11.1* /u/lib » ldd `which dolphin`         linux-vdso.so.1 (0x00007fffdb426000) ...         libQt5Core.so.5 =&gt; not found /u/lib » pacman -Qk qt5-base qt5-base: 3546 total files, 0 missing files   Note that other GUI apps work fine - anything compiled against GTK, including Firefox, work perfectly. I just have to set the $DISPLAY variable. Not setting the var doesn't help, as you'd expect.  Any ideas? Thanks! How would I implement the quantum oracle in Deutsch's algorithm? a) There's a radially symmetric charge density $\rho(r)$ centered around the origin. Determine the dipolemoment of that charge density. b) Let $\rho(r)$ be an arbitrary charge density now. Under what circumstances does the dipole moment of the displaced charge density  $\rho '(\vec{r}) = \rho (\vec{r}-\vec{b})$ differ from the one not displaced at all.  Here were my ideas so far: a) Just thinking about the situation it has to be zero, right? I mean, since there's no real dipole. But how do I show that mathematically? I was thinking of just going like this (it may be wrong): Let the charge density be $\rho (r)=kr$, then we can get the charge q by integrating: $$q=4\pi \int_0^R kr\cdot r^2dr=\pi k R^4$$ I'm looking at the charge distribution as a spherical electron cloud with radius $R$. Then, since $p=qd$ and $d$ is zero because there are no two different charges the dipole moment is zero. Is that sufficient as an answer? b) I don't know how to approach this one. My guess is that its dipole moment is also zero because we're only looking at a displacement here. Anyone got any idea? I would appreciate any advice on this. I have been thinking about ways of teaching electronics and I'm wondering if the following is true...  For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?  So, all other things being equal, will the electrons coming out of a 4 volt battery will have twice the velocity of electrons coming out of a 1 volt battery because of ${1 \over 2}mv^2$?  This would seem to imply that the voltage drop across different parts of the circuit is essentially a comparison of the electron speeds at those points.  Furthermore doesn't this imply that the electrons are nearly halted by the time they reach the positive terminal of the battery? This makes intuitive sense to me, it seems pedagogically sound, and it provides an explanation for how electrons "know" to give up their energy across the circuit -- electrostatic repulsion communicates later resistances to earlier parts of the circuit.  How much of the above is correct? The one things that makes me suspicious is that, in this picture, electron density at the positive terminal of the battery is very high, and repulsive forces would prohibit this. On the other hand, if the Joules per Coulomb are not coming from kinetic energy, then what? The field? But wouldn't that just dissipate as light?  Something is wrong but I don't know what. Why Was It 'Essential' That Voldemort Kill Harry Potter?