Show that if $f: (a,\infty) \rightarrow \mathbb R$ such that $$\lim_{x\to \infty} xf(x) = L$$ where $L \in \mathbb R, $  then  $$ \lim_{x\to \infty} f(x) = 0.   $$
Let $l \in\mathbb{R} $ and $ f:(0,\infty) \to \Bbb R $ be a function such that $ \lim_{x\to \infty} xf(x)=l $ .  Prove that  $ \lim_{x\to \infty} f(x)=0 $.  Any help would be appreciated - I found this difficult to prove
Delay jquery hover event? I would like to delay a hover event in jquery.  I am reading from a file when user hovers over a link or label.  I don't want this event to occur immediately in case the user is just moving the mouse across the screen.  Is there a way to delay the event from firing?  Thank you.  Example code:  $(function() {     $('#container a').hover(function() {         $('&lt;div id="fileinfo" /&gt;').load('ReadTextFileX.aspx',             {filename:'file.txt'},             function() {                 $(this).appendTo('#info');             }          );     },         function() { $('#info').remove(); }     }); });   UPDATE: (1/14/09) After adding the HoverIntent plugin the above code was changed to the following to implement it.  Very simple to implement.  $(function() {     hiConfig = {         sensitivity: 3, // number = sensitivity threshold (must be 1 or higher)         interval: 200, // number = milliseconds for onMouseOver polling interval         timeout: 200, // number = milliseconds delay before onMouseOut         over: function() {             $('&lt;div id="fileinfo" /&gt;').load('ReadTextFileX.aspx', {filename:'file.txt'},                 function() {                    $(this).appendTo('#info');                 }              );         }, // function = onMouseOver callback (REQUIRED)         out: function() { $('#info').remove();  } // function = onMouseOut callback (REQUIRED)     }     $('#container a').hoverIntent(hiConfig) } 90s (or earlier) movie with experimental, invisible, teleporting, time-traveling ship How to add custom Modules in magento Does using links to other pages dilute the linking page's link juice/page rank? There are lot's of questions here about passing link juice to other pages, including .  I've read several of them over the months, but I always come away wondering the same question.  Does linking to another page, dilute or burn some of the link juice the linking page has already earned?  For example, if my current page has for example, a link juice of 10, and I link to another page, it passes a certain amount of juice (I'll use 90% as an example) of the linking page's juice, giving the page being linked to a .9 boost in it's own link juice.  However, does that mean the linking page's page's remaining juice is now less than 10?  While some of the questions would imply this to be the case, it seems counter intuitive as no one would ever want to link to any page outside of their own content for fear of losing page rank. Uniform vs discrete Gaussian sampling in Ring learning with errors The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious. I am a grad student developing an interest for statistics. I like the material over-all, but I sometimes have a hard time thinking about applications to real life. Specifically, my question is about commonly used statistical distributions (normal - beta- gamma etc.). I guess for some cases I get the particular properties that make the distribution quite nice - memoryless property of exponential for example. But for many other cases, I don't have an intuition about both the importance and application areas of the common distributions that we see in textbooks.   There are probably a lot of good sources addressing my concerns, I would be glad if you could share those. I would be a lot more motivated into the material if I could associate it with real-life examples. iBooks library like effect The basic 6T structure used for storing data is same as one used in "Positive Feedback Differential Voltage Sense Amplifier", then how come while the data is stored in SRAM memory cell it doesn't get amplified? Why could the pictures and paintings in Hogwarts talk and the photos of Harry's parents couldn't? Code examples, like in the TikZ/PGF manual I like very much the 'side-by-side' code examples in the TikZ/PGF manual...    ... and I would like to use the same thing in a manual I'm writing. I have found the source code for the codeexample environment that the TikZ/PGF manual uses for this purpose (specifically, it's located at /usr/local/texlive/2012/texmf-dist/doc/generic/pgf/macros/pgfmanual-en-macros.tex).  My (probably rather basic) question is: how can I use the same codeexample environment in my own documents? I ask because I tried compiling the example in the related question below, but the file couldn't be found. $x^3-9=y^2$ find integral solutions Find all integral solutions  $x^3-9=y^2$  I tried many times but still no idea how to solve it. I will be grateful for any help. DNS subdomain (child) NS records If we have example.com we would setup things something like this: $ORIGIN example.com. @  1D  IN  SOA ns1.example.com. hostmaster.example.com. (                   2002022401 ; serial                   3H ; refresh                   15 ; retry                   1w ; expire                   3h ; nxdomain ttl                  )           IN  NS     ns1.example.com.           IN  NS     ns2.example.com.  since in the parent zone .com. there is already NS record for the example.com. child zone and A glue record for the ns1.example.com. for what purpose do we state NS records again in the child zone? and also, do we need to state A record for the name servers in the child zone itself? Can creatures without hands (and as a consequence fingers) use rings? Pretty much what it says on a label. I have a monster for my next adventure that has a pretty interesting ring in his treasure. Since it is intelligent, and would be able to use the ring, I don't see why the ring should be lying there in a chest waiting to be looted.  Monster would use the ring, if it only had hands or even fingers. Hold a second, are fingers even necessary in 5e?  For reference, assume it only has grasping / manipulating tentacles like a  (although it is not one); Said tentacles are very flexible, capable of fine manipulation, and have varying thickness. Google Apps Script Javascript compatibility error I have successfully created . Handy to avoid multiple calls to GAS services whit ordinary getRange calls.  While the code perfectly works in javascript, it throws an error in Google Apps Script upon saving.  The offending code is:  return matrix.slice(startRow, endRow + 1).map(i =&gt; i.slice(startCol, endCol + 1))   GAS seems rejecting code in the .map function. I'm looking to write a script that will find and replace any characters other than [^A-Za-z0-9._-] in a specific file type (in this case .avi) with an underscore _. I want this to exec recursively from the current path. I'm using Ubuntu.  I do not want or need to change any folder names as I have control over those at the time of creation. Here's what I have now after referencing the following link:    find . -depth -exec rename 's!([^/]*\Z)/[^A-Za-z0-9._-]/_/g' *.avi {} +   I'm clearly missing something. Please help. :) Can a Kazakh national transit in Dubai without a visa? Why is a stock that pays a dividend preferrable to one that doesn't? I understand that a dividend is a way for a company to return money to shareholders. But I'm not quite understanding why it is preferable to capital appreciation and putting the money to work increasing the earnings and revenue of the company. I get that people who buy stocks to get income from dividends, but the decrease in the price of a stock that correlates directly to the amount of the dividend seems to me like it's a zero-sum game.  Part of me thinks that I just fall into the Warren Buffet camp of thought that the money can be spent more wisely in house. But the other part feels like the desire for a dividend is to reinvest and add shares at a lower price (the price the stock goes to ex-dividend).   What am I missing here? What is the mechanism by which magnetic fields do work? I've seen some conflicted answers to this question in texts and on the web, in the case of a dipole, for example.   Do magnetic fields do work directly, or is it their induced electric fields that do work? In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$ Let $R$ be a principal ideal domain, and $a, b \in R$ with $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$, and $\gcd(m, n) = 1$. I now want to show that we then already have $a = b$.   I think the statement is easily seen to be true if $a = 0$ (or $b = 0$), since if $a = 0$ and $a^m = b^m$, then $a = b$ since $R$ doesn't contain any nontrivial zero divisors, so $b ≠ 0$ already implicates $b^m = b \cdot b^{m-1} ≠ 0$.  I don't really know how to prove the statement for the general case, where I can't directly apply the fact that $R$ doesn't have any zero divisors except $0$. I most likely have to use the fact at some point that $gcd(a, b) = 1$, but I'm not sure how. Icons of Meta.SE and Stack Apps have gone missing