If using L'Hopital's rule to show $\lim\limits_{x\rightarrow0}\frac{\sin(x)}{x} = 1$ is circular reasoning, then how do we justify it still applies?
How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule?
Why does my Mac OS X 10.6 kernel run in 32-bit mode? I have a MacBook Pro (5,1) with 4GB of memory running 10A432 - but it's running the 32-bit kernel.  Any ideas why it wouldn't be running in 64-bit mode, and how I can get it to do so? How to acknowledge a deceased advisor’s contributions to a paper? One of my advisors suddenly passed away while I was in graduate school.  We had some discussions and ideas about future publications, but he passed away before any of the work was completed.  When the work was finally completed and published, I and my co-authors were therefore presented with an ethical dilemma about how best to acknowledge his contributions to the ideas behind the paper.  Should we list him as a co-author?  Put him in the acknowledgements?  Listing him as an author would give credit for the original idea, however, we would have no way of knowing if he actually approved of—and would want his name attached to—our methods and writing.  In the end my co-authors and I decided to list him as a co-author with a footnote stating that he passed away before publication.  I’m interested to hear from others who have been in similar situations and/or suggestions on what constitutes “co-authorship” when one of one’s collaborators passes away before the publication or work is complete. chkrootkit shows "tcpd" as INFECTED. Is it a false positive? The Fourier transform of a "comb function" is a comb function? Let $f(x) = \sum_{n=-\infty}^{\infty} \delta(x - n)$, where $\delta$ is the Dirac delta function. This function $f$ (a "comb function") is important in signal processing because evenly sampling a function $g$ can be viewed as multiplying $g$ pointwise with $f$. This idea is the first step towards understanding how to approximate the Fourier transform of $g$, given evenly spaced samples of $g$. The next step is to note that $\hat{gf} = \hat{g}*\hat{f}$, where $\hat{f}$ is the Fourier transform of $f$ and $*$ denotes convolution.  Is it true that $\hat f$ is also a comb function? If that statement isn't quite right, what is the correct statement?  I would like to see: 1) an informal, intuitive, nonrigorous but easy derivation of the Fourier transform of $f$ . 2) A rigorous version of the same calculation.  (How would you even define $f$ rigorously, is it a distribution?)  (I'd also be interested in recommendations of math textbooks that cover this topic, including the Nyquist sampling theorem, even if it's only an exercise or series of exercises in an analysis textbook.) Using Squeeze Theorem to show the following limits Consider a function f (x) defined on R satisfying:   for all x distinct 0.  Calculate: lim f(x) with x→∞ and limf(x) with x→−∞   Issue information:   if x → ∞ then we can consider | x | = x  if x → -∞ then we can consider | x | = -x  I can use x² = | x |? The series had three books.  A universe of Greek gods/goddesses, set in modern time.    The gods steal Titans and force them to assume the form of infants/children, and raise them as mundanes in a boarding school type situation.  The protagonist is a young woman who discovers she can travel through the fourth and fifth dimensions as her power.  Her companions: a fey, can travel by unseen ways to various locations; a boy-shaman raised under the traditions of science, can cause anything he wants to happen simply by wishing it and tries to recreate the Michelson–Morley experiment. She ultimately discovers the true nature of the universe she is in and that the deities all follow some sort of rock-papers-scissors system where each deity has a corresponding type which he is weak to, and strong against.  What is the name of this series? What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? SQL Server Stored Procedure Update not working with ASP.Net C# GridView RowUpdating Lines too long due to text inside custom command not being hyphenated I'm having some problems with lines that turn out to be too long. There are correct hyphenation rules for the word but they aren't being applied.  Let me first give the sentence that is too long:  An isomorphism that maps a graph to itself is called an \concept{automorphism}.   The command concept is defined as follows  \newcommand{\concept}[1]{\index{#1}\marginpar{\raggedright\textbf{#1}}\textbf{#1}}   The problem seems to be that words inside \concept aren't being hyphenated. How can I fix this? If I type   An isomorphism that maps a graph to itself is called an \textbf{automorphism}.   then the word is hyphenated perfectly. So I would now like to have the command \concept behave the same way, i.e. allow concepts to be hyphenated according to the standard rules. Find generator of multiplicative group of $\mathbb{F}_{27}$ I have a finite field $\mathbb{F}_{27}$. I need to find an element of order 13. I know that multiplicative group of this field is cyclic with order 26. So I want to find a generator $g$ of this cyclic group, then $g^2$ will have order 13. Also I know that all elements of this field can be represented by polynomials from $\mathbb{F}_3[t]$, where t is root of irreducible polynom $x^3-x-1$.  Probably we can find generator of $\mathbb{Z}_{26}$ and than build an isomorphism to $\mathbb{F}_{27}\setminus 0$.  Anyway I don't know were to start.  Thanks! Material setings in properties window is missing I have added a sphere and I was able to add material and give it a color. Now when I add a second sphere (separate object), the material panel is just blank. How can mixed content compromise an entire HTTPS session? How can I access a JavaScript object which has spaces in the object's key? Looking for old book, SF, artificial world, different zones for different species that visitors "inhabit" How to send an SMS in iOS 10 Map not loading when using getLegend and showing error for getView() I was able to run my application on using Legend earlier but now I want to update my Legend when scrolling on the map. So I tried to used this code  . And I got the gist of how he is able to do that but when I am trying the same I keeping getting the error var graphicUrl = wmsSource.getLegendUrl(currentResolution);  I think this is mainly due to the wmsSource not declare. But I tried to initialize  seems wmsSource in TileWMS but does not seems to work. var getExtent = function () {     return map.getView().calculateExtent(); } var getProjectionCode = function () {     return map.getView().getProjection().getCode(); }  var updateLegend = function() {     var currentResolution = map.getView().getResolution();     var graphicUrl = wmsSource.getLegendUrl(currentResolution);        const crs = `&amp;CRS=${getProjectionCode()}`;     const bbox = `&amp;BBOX=${getExtent().join(',')}`;     const height = getHeight(getExtent());     const width = getWidth(getExtent());     const heightAndWidth = `&amp;srcwidth=${height}&amp;srcheight=${width}` ;     const legenOptions = &quot;&amp;legend_options=countMatched:true;fontAntiAliasing:true;hideEmptyRules:true;forceLabels:on&quot; ;        var img = document.getElementById('legend');     img.src = `${graphicUrl}${crs}${bbox}${heightAndWidth}${legenOptions}`; }; //Updated Legend End   //View var myview = new ol.View({     center : [8717490.201868,2788422.791843],     zoom : 4 })  var firstSel = document.getElementById(&quot;City&quot;); var secSel = document.getElementById(&quot;Sector&quot;);  //var LayerArray = [baseLayer, firstLayer]  //Map var map = new ol.Map({       target : 'mymap',       view: myview,       layers: [         new ol.layer.Tile({             source : new ol.source.OSM({                 attributions: 'Surveyor Allpication'             })         }),         new ol.layer.Tile({             source : new ol.source.TileWMS({                url: 'http://localhost:8089/geoserver/India/wms',                params: {'LAYERS': 'India:India', 'TILED': true,'TRANSPARENT':true},                ratio: 1,                serverType: 'geoserver',                transition: 0,             })         })       ] }) updateLegend();   This here is my JS &lt;div class=&quot;innerMap&quot;&gt;     &lt;div class=&quot;map&quot; id=&quot;mymap&quot;&gt;&lt;/div&gt;         Legend:     &lt;div&gt;&lt;img id=&quot;legend&quot;/&gt;&lt;/div&gt;     &lt;!--    &lt;div&gt;&lt;img src=&quot;http://localhost:8089/geoserver/wms?REQUEST=GetLegendGraphic&amp;VERSION=1.0.0&amp;FORMAT=image/png&amp;WIDTH=20&amp;HEIGHT=20&amp;LAYER=India%3AIndia&amp;RULE=Agra&quot; /&gt;%Name&lt;/div&gt; --&gt; &lt;!--    &lt;div&gt;&lt;img src=&quot;http://localhost:8089/geoserver/India/wms?SERVICE=WMS&amp;VERSION=1.1.1&amp;REQUEST=GetLegendGraphic&amp;FORMAT=image/png&amp;WIDTH=20&amp;HEIGHT=20&amp;LAYER=India%3AIndia&amp;SRS=EPSG%3A4326&amp;STYLES=&amp;SRCWIDTH=742&amp;SRCHEIGHT=768&amp;BBOX=75.4541015625%2C14.21630859375%2C91.7578125%2C31.09130859375&amp;LEGEND_OPTIONS=countMatched:true;fontAntiAliasing:true;hideEmptyRules:true&quot; /&gt; &lt;/div&gt;  --&gt;     &lt;/div&gt;  This is my HTML. How do I make this work? Can you create an illusion of empty space? I've been thinking of various ways for a crafty Illusionist/Rogue to 'hide in plain sight' by essentially creating the cover that they want to hide behind.  Ideas include things like:   Create a wall that is 2.5ft closer than it really is and hide in said space Create some innocuous barrier or container and hide within it Create a 'sheet' that essentially has what an empty room looks like 'painted' on it, so someone walking past an open door would see an empty room.   From different discussions I've had, these all should work, but the one lingering one I have is, what if my illusion is simply 'empty space'?  If I were to create an illusion of empty space and then place myself in it, would I be essentially invisible?  What if I created an illusion that was partially-empty space, like a 5x5 cube of swiss cheese? Would it seem like there was a swiss cheese cube formed around my person or could my 'fake-empty-space-image' fill the holes?  Would my 'empty space' illusion simply need to work like the aforementioned 'sheet of what I expect you to see' in order to function (and thus most likely only work from limited viewing angles)? How to avoid the wrinkle at the poles of the UV sphere? How can I show that the order of an element modulo $n$ divides $\phi(n)$?  I know that if $a$ and $n$ are relatively prime, then the least positive integer $x$ such that $a^x\equiv1\pmod n$ is its order modulo $n$. I also know that, by Euler's theorem, $a^{\phi(n)}\equiv1\pmod n$. Therefore, it must be the case that $x\leq\phi(n)$.  However, all that I am left to do is to show that $kx=\phi(n)$, for some integer $k$. Do you guys have an idea on how to do this? Thanks in advance! Meaning of hash symbol in pin name I'm reading the datasheet of a PSRAM (available ). In the top-right hand corner all the pins are listed. Some of the pin names end with a hash, e.g. LB#, OE#.  What does the # notation mean for pins?