I know, how to calculate $$ \lim_{x\to0}\frac{\cos x-1}{x^2} $$ without differential calculus. Calculating $$ \lim_{x\to0}\frac{\sin x-x}{x^3} $$ using de l'Hospital's rule or Taylor expansion is also easy.  Is there a method to calculate the previous limit without de l'Hospital's rule or stronger tools?
Is it always possible to find the limit of a function without using  or ?  For example,   $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$  $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$  $$\lim_{x\to0}\frac{\ln(1+x)-x}{x^2}$$  $$\lim_{x\to0}\frac{e^x-x-1}{x^2}$$  $$\lim_{x\to0}\frac{\sin^{-1}x-x}{x^3}$$  $$\lim_{x\to0}\frac{\tan^{-1}x-x}{x^3}$$
If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.   I want to use induction to prove this:  It is trivial when $G=p(n=1)$, assume $G=p^n$, $N$ is a normal subgroup in $G$ with order $p$. Since $G$ is p-subgroup, $G$ has a normal subgroup $H$ with order $p^{n-1}$,according to assumption, $N \subset C(H)$. But I don't know how to go on, is $C(H)=C(G)$?Can induction work in this case? Tenure and tenure denial rates by universities Is there a way or website to find the tenure and tenure denial rates by universities and departments? I heard from some faculty that the tenure rate at X university is Y%, so I wonder if there is a way to find the statistics for all universities. How to perform the modular reduce of Rijndael's finite field I am trying to understand how to calculate the modular reduction of Rijndael's finite field.  The example on  page says that {53} • {CA} = {01}, because (11111101111110 mod 100011011) = {3F7E mod 11B} = {01} = 1.  I am able to get to the point where I have 3F7E mod 11B, but cant seem to figure out how the reduction works.        11111101111110 (mod) 100011011      ^100011011             1110000011110       ^100011011             110110101110        ^100011011             10101110110         ^100011011             0100011010            ^100011011 &lt;-- here, right shifted two places?             00000001   From this example, why does it right shift two places at one point?  I also found an example that 0x1b*0xaa equals 0x8c. Trying to solve this myself I get to the point: 0xE0E mod 0x11B   111000001110 ^100011011   11011010110  ^100011011    1010111010   ^100011011     010001100    ^100011011      10010111   This makes it 0x97 The Stack Overflow search shows questions twice. See  for example.  Results include "A: Add Search Engine into MVC Music Store" and "Q: Add Search Engine into MVC Music Store".  Why is search displaying the question twice, and what does "Q" and "A" mean?  Edit:   @Stephen answer :  The search is returning both a question (Q) and an answer (A) to the question, as both posts match your search criteria.  and my another question  If the question have multiple (like 100) answer ,then the answer link(A: Question Link) shown 100 times?  Is it correct functionality ? But i saw only one answer link , not a other answer link .. why ? Solve for $x$: $2^x=4x$ Given that $x$ is a positive integer. By using methods of trial and error as well as plotting two lines: $y=2^x$, $y=4x$ on a graph and find their intersection point, we can easily solve for $x$ which is equal to 4. However, I do not know how to solve this using only equation. Can anyone help me? Fantasy series with Celtic overtones, intermittent stone magic, and ongoing slow magic I checked this series out of the library during college, maybe 15 years ago now.    It was based around 'stone magic' – magical stones that granted their bearers enormous power.  Unfortunately, the power of these stones is intermittent, they'll fade out for a long time then fade back in.  The main characters dealt with a 'master' stone of some sort, the first to restore, last to die.  It's also the key to granting other people access to their own powers – I think seals had magic fish they swallowed, which keyed off the master stone, and in the last book I think the character created magic staves for another race. It also had 'slow magic' – longer to set up, less apparently potent than stone magic, but the slow magic, given time to work it, could still be powerful and dramatic.  It just took time and preparation instead of brute force. In one book, the main character faces off against the source (or something like that) of magic, and is about to win when it's discovered she's pregnant.  The source – a dragon? – isn't allowed to harm anyone who doesn't come to it freely, so the main character is forbidden to finish the trial.  In the sequel, her daughter repeats the trial, and becomes immensely powerful by winning, but there was a cost of some sort, something that transformed her A lot of the naming made me think it had a Celtic root. How do I cash a USD cheque in Europe, quickly? I'll receive a USD cheque shortly. I prefered payment by money transfer, but all attempts to do so - i.e. SWIFT/IBAN, Payoneer, transferwise - failed. Now I will receive a cheque by mail.  I contacted my bank and they say that they can cash a cheque for me, given a (prob hefty) fee. The bank employee was so kind to warn me, that in case of a cheque the funds are cashed, but reserved until the legal reimbursement period ends, which apparently is between 3 and 6 months.   Is there really no quicker way to receive payments by cheque and have access to the funds quickly? There is a sentence:     No sooner had he sat down than he fell asleep.   I just do not understand, is this an inversion? And if so, I still do not understand the sentence. having bracket in front of an item What is the angle at which water is splashed when a vehicle tyre rides in water on the road? I would like to know the following:   What is the angle at which water gets splashed when I ride my vehicle through a water on the road? How does angle of water varies with speed? What is the relation between the distance the water goes with the speed of the vehicle?   Thanks! Finding the missing digits of $23!$ Film about some dude who fell in love with a star (a literal stellar object) Yes, the plot is pretty weird. There is this "star" (whose incarnation is a woman dressed in white) and some random guy who falls in love with her. There's the arc of "if you cross a specific border you'll die" which applies to our lady "star". Well, I also remember that they have some kind of a tubular case, and when one opens it, lightning files out.  I watched the film somewhere around 2010, it's in English and when I try to remember it, I get the vibe of a mediocre Hollywood fantasy romance. Passing arguments forward to another javascript function How can Linux/Ubuntu be installed safely on ChromeBook? Is installing Ubuntu on my Chromebook potentially dangerous to anything on my Chromebook? I got it as a gift, so I don't want to destroy anything that is important on the laptop. Also, which is better for being able to safely install Ubuntu: ChrUbuntu or Crouton? I'm using the Samsung Chromebook 3 (as described by the quick-start manual). Does that make any other differences as well? Loading OpenStreetMap data in QGIS? In QGIS 1.8 I used the openstreetmap plugin to request data from the server, or open a .osm XML file, which produced 3 layers (polygons, polylines and points) in QGIS.  This was great (give or take the 64bit bug which highly affects the freshly mapped area I work on), but how do I do that in QGIS 2.x?   I can't find any updated documentation about it.  In the vector menu, I've tried importing from osm servers, I get a .osm file, but then I'm stuck. There's a menu option for loading an XML file, but it produces a spatialite db, which apparently only has non-geometry tables when I try to add it as a spatialite layer. If I open the attribute table, I can see node ids, ways and so on, but I can't figure out how to actually make use of them.  I've also tried adding a vector layer from a .osm XML file. I get a gpsbabel window that shows nothing, then finally an invalid data source message in QGIS. Notation for infinite product in reverse order This question is related to notation of infinite product.   We know that, $$ \prod_{i=1}^{\infty}x_{i}=x_{1}x_{2}x_{3}\cdots $$  How do I denote $$     \cdots x_{3}x_{2}x_{1}  ?    $$  One approach could be  $$ \prod_{i=\infty}^{1}x_{i}=\cdots x_{3}x_{2}x_{1} $$  I need to use this expression in a bigger expression so I need a good notation for this. Thank you in advance for your help. How to get glare on "Quick Smoke" (Fire & Smoke explosion) Replacing NAs with latest non-NA value Stack Exchange sites need a universal clock that is visible! Since the Stack Exchange sites run on one clock, and a clock that is not common to most of us (such as those of us in the US), can we display a clock with the site time (UTC) displayed somewhere on every page, such as at the bottom with the other text?  That way we are more aware of when the days turn over, and ecetera...  close votes, delete votes, flags, and normal votes would also benefit from this as well (thanks @Grace Note!)...    What do you guys think? yea or nay? thoughts? Currentsection TOC in scrartcl? I like to get a toc at beginning of some of my sections, like in beamer where I can use \tableofcontents[currentsection] is there a possibility in scrartcl?  Something like this:  \documentclass[10pt,a4paper]{scrartcl} \usepackage[ngerman]{babel}  \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \tableofcontents %complete  \section{First}  \section{Second} \tableofcontents %just subsection and subsubsection \subsection{SecondSub} \subsubsection{please} \subsection{this} \subsection{SecondSub2} \subsection{parts}  \section{Third} \subsection{another}  \end{document}   Looks like:  Inhaltsverzeichnis %complete  1 First  2 Second 2.1 SecondSub.......................2 2.1.1 please........................2 2.1.2 this..........................2 2.2. SecondSub2.....................3 2.2.1 parts.........................4  3 Third 3.1 another ........................5   1 First  2 Second  Inhaltsverzeichnis %just subsection and subsubsection 2.1 SecondSub.......................2  2.1.1 please........................2  2.1.2 this..........................2 2.2. SecondSub2.....................3  2.2.1 parts.........................4  2.1 SecondSub  2.1.1 please   2.1.2 this  2.2. SecondSub2  2.2.1 parts   3 Third  3.1 another