$$\lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2}$$  This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or Squeeze Theorem. For each attempt, I expected at least one of  the well-known limits $\lim_{x \to 0} \frac{\sin{x}}{x}$ or $\lim_{x \to 0} \frac{\cos{x} - 1}{x}$ to come into use. They frequently did in my attempts but did not go anywhere. I have also been experimented substitutions but in vain.  How can this be solved without using L'Hospital's Rule?
I am a kid trying to teach myself Calculus in order to prepare for next year.   I have the expression  $$\lim_{x\to 2}\frac{\cos(\frac \pi x)}{x-2} $$  There is a hint that says to substitute t for $(\frac \pi2 - \frac \pi x)$ and WolframAlpha evaluates this expression as $\frac \pi4$. However, I got the answer of $1$. Can someone clarify the steps to solving this problem.
is $f$ constant or dilation $f:\mathbb{R}^3\to\mathbb{R}, f(rx)=r^{\alpha}f(x)$ for some $\alpha&gt;0$, $\forall x\in\mathbb{R}$, for any $r\in\mathbb{R}$, could any one tell me which of the following is true?  $1.$ If $f(x)=f(y)$ whenever $||x||=||y||=\beta&gt;0$, then $f(x)=\beta ||x||^{\alpha}$  $2.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)= ||x||^{\alpha}$  $3.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)=c||x||^{\alpha}$ for some constant $c$  $4.$ If $f(x)=f(y)$ whenever $||x||=||y||$, then $f(x)$ must be a constant function. Thank you. What is the simplest organism that was observed to learn: change its behavior permanently in response to some event/stimuli in a way that this change persists even if the event does not happen again? Any point to downvoting already closed questions? That question of mine was closed a long time ago (relatively, to me at least).  However, from time to time I still get downvotes for it.  That got me thinking: is there any reason to still downvote closed questions? Rewrite rules goes away Black is Back! New update with Achievement shows black backgrounds with black text Update: This is fixed on version 1.0.43 as per the accepted answer below.  Already saw ,  and . They are old and seem to be about previous versions.  I just updated to the version 1.0.41 of the Android app that brings the new feature - Achievements. Sadly, I think the stress of World Cup has made you guys not do a good job with it. It's buggy.  When I open my Achievements panel and tap on any post, I get this:    After I get this, it persists throughout the rest of the app. For example, if I go to the feed and open another question, I get this too.  Detailed reproduction steps are available in AD7six's answer.  My Device: Samsung Galaxy SIII LTE GT-I9305 Android: Stock ROM 4.3 Jelly Bean  Other Devices: - Device model, Android version, Reported by - HTC Desire, 4.2.2, xd6_ - Huawei Ascend, 4.2, Manu343726 - LG Optimus F3, 4.1.2, Rocket Hazmat - LG Optimus G, 4.3, Canadian Luke - Samsung Galaxy Nexus (Sprint), 4.3, Chrono Kitsune - Samsung Galaxy Note II, 4.3, Katja Christiansen - Samsung Galaxy SII GT-I9100, 4.1.2, AD7six - Sharp SH837W, 4.1.2, timrau - Sony Xperia SP, 4.3, pepoluan - Sony Xperia Z1, 4.3, user2313067 - Unknown, 4.2.1, Deer Hunter What does "soft" in "soft symmetry breaking" mean? Opacity Mask With Multiple Images (Illustrator CS5) I am trying to do an opacity mask and it is driving me crazy.  Bascially I have a layer of ice. It's four different pictures of ice overlapping each other so it looks like a whole bunch of ice. I want to have it fade in from both sides (left and right) until it is completely opaque in the middle.  I've select all of the pictures and chosen 'Make Opacity Mask' from the Transparency Panel and it disappears but then I don't know how to manipulate the opacity.   In the Transparency panel it doesn't show a picture of what I'm working with so I can't simply click next to it to create the opacity mask like I've seen in tutorials.  I've also tried to overlay a rectangle over the objects, choose them all then try 'Make Opacity Mask' but that didn't work either. Any ideas? Thanks. How can I produce an encrypted Swap Partition for 14.04 ? All the advices I saw here are for versions &lt;14.04 and do not work for me.  I tried a lot but nothing lasted longer than a reboot.  On installation I decided to encrypt my home and 9 GiB partition for swap but this swap was never used by Kubuntu. Gparted finds an "unknown filesystem".  I many times formated it as swap, edited fstab and crypttab and used all the tricks mentioned in different answers here. The often mentioned PunyGeek letter does not work for me probably because /etc/initramfs-tools/conf.d/resume does not even exist on my system. AppleScript to choose item on web page Is it possible with AppleScript to choose a clickable element in a web page and click it and if so, can you give me an example? Do "Best by: XX-XX-XX" foods have an expiration date? How long will foods that have "Best by: XX-XX-XX" on the product last past that date?  I have tried looking online, and all I can find is anecdotal evidence, which isn't very assuring for me since there seems to be no specific "x food will expire y weeks after its best by date".  Is there a rule of thumb or some guide out there that can tell me how long a "best by" food will last? Or is it safe to assume that they never expire, since it's not actually labelled on the product? Implementing support for nested transactions using JDBC 3.0 our legacy application uses JDBC 3.0. it supports transactions by implementing its own transaction manager that is capable of returning the same JDBC connection for each thread. The problem I recently discovered is that it doesn't support nested transactions: if a transaction is started inside an another transaction, then every SQLs running in the context of the inner transaction will be executed using the same db connection, and when it's commited or rolled back it will automatically commit or roll back all the changes starting from the outer transaction.  As far as I understand I can implement the support for nested transaction using JDBC 3.0 savepoints: whenever a nested transaction is started, I can set a new savepoint for the current connection. Afterward, if the nested transaction is rolled back, I will just rollback to this savepoint. If, on the other hand, it is commited, I will just do nothing. Only the commit of the most outer transaction will save the changes to the db.  Is this correct? Does this approach have any flaws? If yes, what are my possibilities?  Thanks. I am intermediate in PHP. To polish my skills I start learning Drupal 7. While learning Drupal architecture concepts, the terms hooks and bootstrapping puzzled me a lot. I read the "Pro Drupal development" book, and some documentation on drupal.org, but it is so advanced for me to learn how hooks work in Drupal to display the web page.  Can anyone tell me what hooks are in simple words? A concrete example of involution in group theory I am reading the textbook "Introduction to Modern Algebra, Joyce 2017" and in the Cyclic groups and subgroups section, there is a following sentence about involution.     An involution $a$ is an element of a group which is its own inverse, $a^{-1} = a$. Clearly, the order of an involution $a$ is $2$ unless $a = 1$, in which case the order of $a$ is $1$.    I was trying to come up with a concrete example, but I am having difficulties. Maybe when $a = \frac{1}{2}, \frac{1}{2}^2 = 1$? I also searched the  for a concrete example, but wasn't able to find anything.  Can someone provide an example with numbers for me please? I want to understand why the author writes "clearly" that the order of an involution $a$ is $2$. Thank you I think some good words would be conflicted, or confused. As an example of context, think of someone who believes in freedom of speech, but, at the same time, they would support banning comedians who make jokes about certain issues that person cares about. The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? Can somebody help me understanding the fouries series coefficients?  I know that if we have:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$  And because of the $\sin(a+b)$ formula: $$ f(t) = \sin(2\pi nt + Ph) = \sin(2\pi nt )\cos(Ph_n) + \cos(2\pi nt )\sin(Ph_n)$$ Then:  $$\sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (A_n\sin(2\pi nt )\cos(Ph_n) + A_n\cos(2\pi nt )\sin(Ph_n))$$ And by definition: $$a_n = A_n\cos(Ph_n)$$ $$b_n = A_n\sin(Ph_n)$$ Then:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (a_n\sin(2\pi nt ) + b_n\cos(2\pi nt ))$$ Where does the $A_n$ in the first formula come from? Could somebody explain me where does the $\frac{a_0}{2}$ in the formula below come from? $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos(nx) + b_n \sin(nx))$$  And the most important:  How can I proof the formulas for the $a_n$ and $b_n$ coefficients?  PLEASE, I'm searchinf for these answers for DAYS. I want to create a wave made out of particles/dots. The wave should look like this:  I tried to achieve this effect by adding an ocean modifier to a plane. Then I added a particle system to the plane but now I don't know how to go on. How can I make the particles lay on the 'ocean' and move when the ocean is moving? PS: I know there is a way to do this with a material in the Blender Render. But firstly I want to create the whole thing in Cycles and secondly I want to have particles. Does double negation distribute over implication intuitionistically? How many regenerations does the Doctor have after The Time of the Doctor? Upon realizing it is a sarcophagus, he decides to sacrifice himself (I think by disconnecting his air) and exist in it for eternity along with the original creature. It may have been in an old Omni magazine, or some other periodical. I thought it may have been in the anthology "Dead Astronauts" but I did not find the story in there.