$\lim_{x \to x_0}{\frac{\arcsin x-\arcsin x_0}{x-x_0}}=?$
question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$
Why should a CTE start with a semi-colon? I was just looking at a post on  where Aaron Bertrand proposes using a CTE instead of a numbers table, which is an elegant way of performing the task at hand.  My question is, why does the first line of the CTE begin with a semi-colon?    ;WITH n AS (SELECT TOP (10000) n FROM    (SELECT n = ROW_NUMBER() OVER     (ORDER BY s1.[object_id])     FROM sys.all_objects AS s1     CROSS JOIN sys.all_objects AS s2   ) AS x ORDER BY n ) SELECT n FROM n ORDER BY n; -- look ma, no gaps!   Is this to ensure the WITH statement does not get parsed into a previous SELECT or something?  I see nothing in SQL Server 2005 BOL about using a semi-colon prior to the WITH. Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation What kind of node is this? Is centripetal acceleration mandatory for circular motion? Most of the recipes I'm using (mainly Modernist; stocks, caramelized vegetables, confit) suggest cooking at 15psi. My pressure cooker has two settings 12psi and 7psi. I've always had success just using 12psi in recipes where it says 15psi. However recently the 12psi mode has broken on my PC. I can now only cook at 7psi. How can I convert my recipes to use this lower pressure?  For example I would cook a chicken stock at 12psi for 90mins, I presume at 7psi the cooking time would need to be longer but by how much? PostgreSQL execute a default query when no row are returned I need to calculate the average score of a list of userId.  I have a table user_place_score:  (user_id, place_id, score)   Sometimes I do not have the score (no row for (user_id,place_id) in the table) so I need to fetch a "default" score from the table default_place_score:  (place_id, score)   Is is possible in PostgreSQL to do this in one query? For a given list of users and one place_id, select all their scores in the user_place_score and if no row is found then take the score found if the default_place_score table?  Basically I want to avoid doing this kind of stuff:  for (Integer userId : userIds) {   score = (query to fetch the score for this userId)   if (score == null)     defaultScore = (query to fetch the default score for this PlaceId)    total = total + defaultScore }  //return the average return total/numberOfUser Why does mathematical convention deal so ineptly with multisets? New questions, Show Recommended, Sort Newest not working Why is it wrong to serial voteup someone you like his answers?  Sometimes I just came to stackoverflow to read certain people answers(there are 4 guys currently) I have their profile bookmarked in my browser and in my free time at office I go to their profile and read their answers and learn a lot from them, but because I'm afraid to be a serial voter, I dont voteup them.   I think there is nothing wrong with that, if there is, please let me know. Can I play Minecraft Java if I only buy the Windows 10 version I do not own Minecraft outside my PS4, I know I can't play either Windows 10 or Java with that purchase. However, my primary computer is Linux and only supports the Java edition, and its past the deadline to get a redeem code for Windows 10 edition if I purchase the Java Edition.    Can I purchase the Windows 10 edition and use that purchase to login and play on the Java edition? Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$ Test the convergence of the series For $a&gt;0$, test the convergence of the infinite series: $$\sum a^{\ln(n)}$$ Which test should be better to use here? I tried the D'Alembert's ratio test. $$\lim_{n\to \infty} {\frac{a^{\ln(n)}}{a^{\ln(n+1)}}}= \lim_{n\to \infty}a^{ln{\frac{n}{n+1}}}=a^0=1$$ So the test failed.  UPDATE: I observed that $${a^{\ln(n)}}=e^{{\ln(a)}{\ln(n)}}={n^{\ln(a)}}={{1\over n^{ln({1\over a})}} }$$Further by test of auxiliary series $$\sum \frac{1}{n^p}$$ we know that criterion for convergence is $p&gt;1$. Here $p=ln({1\over a})$.Hence the series will be convergent when $ln(a)&lt;-1$ i.e. when $0&lt;a&lt;1/e$, since $a&gt;0$ is already given.Finally this solved my question. Get the subdomain from a URL Getting the subdomain from a URL sounds easy at first.  http://www.domain.example   Scan for the first period then return whatever came after the "http://" ...  Then you remember  http://super.duper.domain.example   Oh. So then you think, okay, find the last period, go back a word and get everything before!   Then you remember  http://super.duper.domain.co.uk   And you're back to square one. Anyone have any great ideas besides storing a list of all TLDs? I want to supply constant voltage to my LED's but the supply varies from 3.7V to 4.2V (Li-ion battery). I cannot use 3.3V voltage regulator as it will need at least 4.3V voltage at the input but my battery can supply only 4.2V max. so is there any other way I can supply regulated 3.3V to my LED.   RGB LED = 3.3V, 60mA Battery = 3.7V - 4.2V, 3000mA Number of LEDs = 24 The LED's are connected in parallel Java Enum definition I thought I understood Java generics pretty well, but then I came across the following in java.lang.Enum:  class Enum&lt;E extends Enum&lt;E&gt;&gt;   Could someone explain how to interpret this type parameter? Bonus points for providing other examples of where a similar type parameter could be used. Limit of derivative is zero. Does it imply a limit for f(x)? Are battery packs allowed in hand luggage? When flying, are you allowed to bring battery packs / power banks? I'd like to bring my ACC 10400mAh Power Bank on several flights. Theorem numbering Making a HUD/GUI with OpenGL (LWJGL) Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above).