To be honest, I have to solve the following exercise, but I don't know what trigonometric formulas should I use (if I should), to get to a form where I can use the Taylor series. $$\lim_{x\rightarrow 0} \frac{\sqrt{1+ x\sin(x)} - \sqrt{\cos(2x)}}{ \tan^2(x/2)}$$ I have tried to convert the tan function (using trig formulas) but it didn't work, also I tried to use Taylor series on the $\sin(x), \cos(2x)$ functions.
I need to evaluate the limit without using l'Hopital's rule.  $$\lim_{x \to 0}{\frac{\sqrt{1+x\sin{x}}-\sqrt{\cos{2x}}}{\tan^{2}{\frac{x}{2}}}}$$
What processing is done on RAW files in the camera? Recently I came across a debate about whether it's fair to call RAW 'unprocessed' or 'unaltered' sensor data of a DSLR. As far as I know, the analog sensor data is processed into a digital RAW file using the ISO setting, a process during which the Bayer-sensor data is demosaiced as well. However, those are just the necessary steps for saving a digital file while maintaining maximum quality.  Are those assumptions correct? If so, are there any additional steps that need to be performed between the sensor receiving light and the saving of the RAW file? And can those be reasonably be described as an 'alteration' of the sensor data, or is it fair to say that a RAW file holds the unaltered sensor data? Keeping an updated balance (Accounting) I have been struggling trying to find a way to properly and reliably maintain a balance for accounts in an order/payment system.  Currently, the process consists of three tables. Below example is very simplified but gets the point across.  Table one contains the account ID, name, total paid, total orders, balance etc.  Table two contains payments, date, the account ID (FK) and amount.  Table three contains orders, date, the account ID (FK) and amount.  Calculating the balance is of course as easy as  SELECT SUM(payments.AMOUNT) FROM payments WHERE ID = [account] SELECT SUM(orders.AMOUNT) FROM orders WHERE ID = [account]   Followed by some simple addition/subtraction math:  UPDATE accounts SET BALANCE = [balance] WHERE ID = [account]   All good - the problem is - how do I maintain this over let's assume years of activity and many, many transactions?  Currently the accounts table is updated every single time a transaction is made, meaning the above three statements run every time. There is of course an index on the ID column in all three tables, and everything is fast and neat for now, but I work with less than 100,000 transactions, meaning everything fits into the RAM and basically all I throw at the database gets executed immediately. Over time this will change, of course - so how do I handle this?  Is it better to mark the accounts as "dirty" and process balancing in a batch job later, or can I keep updating the balance real-time even when I reach the point where the orders and payments tables will not fit into the RAM?  Do I need to segment the orders and payments and "lock" everything before a given time, such as before last calculation? (I would rather not). I ask because then I could do a SUM WHERE DATE > [last lock timestamp] and update with UPDATE BALANCE = [balance] + [value for last lock] and prevent the problem from outgrowing resources.  I don't need a running balance for each transaction. That's all I can find results for when I try to research this problem. I just care about the total amount paid, the total amount ordered and the balance of accounts.  Why don't I calculate on the fly when I need the balance, you ask? Because one must be able to query the system and ask "Which of my (let's say) 50.000 accounts have a balance below 0?" - something we cannot do easily if we do not store the balance with every account. Android popup window dismissal How many $m\times n$ tables are there if they are subject to the following two conditions?    In each cell of the table we have $1$ or $-1$. The product of all the cells in any given row and the product of all cells in any given column equals $-1$.    I already know if Parity of $m$ and $n$ are different there is no such a table. what is the answer in general? Multiplying Binomial Terms Proof of Frullani's theorem I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? Convergence of the $\sum_{n=1}^{\infty}n^3b_n^3$ where $b_n=\int_{-\pi}^{\pi}f(t)\sin (nt)\ dt$. Which is correct: "he don't" or "he doesn't"? Which one is correct in a sentence?        He don't    He doesn't      I guess "he doesn't" should be correct because he is third person singular but I've seen some people using do with he. Which one is correct? Conjugate to $\cos z$ is not equal to $\cos z$ conjugate $\overline{\cos(z)}\neq\cos(\overline{z})$ I have a dilemma. I have a task where I'm supposed to show that $|\cos z|^2+|\sin z|^2=1$ if and only if $z\in \mathbb{R} $. In my argument I have that $$\cos z\cdot\overline{\cos z} + \sin z\cdot\overline{\sin z}=\cos z\cdot \cos\overline{z} + \sin z\cdot \sin\overline{z}$$ but my teacher said that $\overline {\cos z} \neq \cos \overline{z}$.  Is this true?  But $$\overline{\cos(z)} = \overline{\frac{e^{iz}+e^{-iz}}{2}}=\frac{e^{-i\overline{z}}+e^{i\overline{z}}}{2}=\cos(\overline{z})$$ Why use the 'ref' keyword when passing an object? If I am passing an object to a method, why should I use the ref keyword? Isn't this the default behaviour anyway?  For example:  class Program {     static void Main(string[] args)     {         TestRef t = new TestRef();         t.Something = "Foo";          DoSomething(t);         Console.WriteLine(t.Something);     }      static public void DoSomething(TestRef t)     {         t.Something = "Bar";     } }   public class TestRef {     public string Something { get; set; } }   The output is "Bar" which means that the object was passed as a reference. Huge close votes review queue on Stack Overflow Today, the size of the close-votes queue is 55k questions and growing. Is that a problem?   Which action could be taken to clean it up? It seems to be growing faster than Stack&nbsp;Overflow members are capable to handle it. When should I use \input vs. \include? There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of? I shall come, not will Native speakers, how can you comment such a sentence:  I shall come, not will  Can I use it for  I will definitely come, not maybe    Adding some information other people asked me about.  How I came to such a question.  Here   I asked about "Shall I win?" pointing that it could have some variants of understanding:  1) Shall I win? = Should I win? (suggestion)  2) Shall I win? = Will I win? (archaic with 1st person)  3) Shall I win? = is it 100% that I will win? (predestination\inevitability)  The excellent guy Mixolydian said either variant was possible, that it didn't matter what person you used "shall" with, it could have this "predestination\inevitability" anyway. Out of this we're getting that nowadays we can use "will" for simple future actions and "shall" for almost predicted (almost 100% surety of happening) future actions.  If I want to say to someone:  Don't be afraid, I will come anyway.  Don't get panicked, I will definitely come  Be sure, I will come in any case  I can use "shall" for it meanining an action into which I will put 100% effort:  I shall come!  And what if my original sentence is  Don't doubt, I will definitely come, not maybe or probably  Can I use  I shall come, not will  pointing that "shall" guarantees me putting all my strengths into it while "will" doesn't give any assuarance that I will do everything to finally come.  As I understood, it wasn't very easy to ask such a question because English was different everywhere - US, UK, Australie, Ireland etc. I heard it was acceptabse to use even "thee\thou" in Ireland or somewhere like that while all other world has already forgotten such words...  P.S. What I got from the answers ias that it was better to forget about the existence of "shall" cause it created so many misunderstandings:D Limit size of Queue in .NET? I have a Queue&lt;T&gt; object that I have initialised to a capacity of 2, but obviously that is just the capacity and it keeps expanding as I add items.  Is there already an object that automatically dequeues an item when the limit is reached, or is the best solution to create my own inherited class? math behind complex numbers C||RLC I have a mathematical problem with the following formula. The goal is to transform this equation to have a seperated real and imaginary part. I have already done this with several other equasions, but for this particualar one I don't know where to start.  I dont expect you to do the complete work of transforming for me, it would already be very nice if you could give me a hind on how you would start.  Thank you in advance!  C||RLC      This is my current progress:    My question is:  How can I get rid of the imaginary part jωC2*R?  When the denominator is completely real I can just split up the formula and have an imaginary and a real part. I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 80/90s UK children's drama series about an "alien" boy with radio weapon I've been trying to track down a live-action children's TV series (or short story perhaps) from the 1980s or 1990s, probably on CBBC or CITV.  A few people I've spoken to IRL have suggested The Boy from Space or Sky, but I'm pretty sure it's neither as a lot of the details don't seem to match.  The story was set on Earth (possibly England or Wales) and was about an "alien" boy who was being pursued by a group of people, though I don't remember who they were.  Somehow he encountered siblings, a girl and a boy, and made friends with them. I think they lived on a farm with their parents, as I remember quite a large barn or other building, and they hid(?) the boy in a car which was inside or next to this building. He also slept in this car with a dog (which might have been a German shepherd).  The "alien" boy had a portable "radio" (all silver, very 1980s) that repelled his pursuers by emitting a high-pitched sound when he turned a dial on it, and he used it when out and about with the other children whenever they encountered a pursuer. I seem to remember lots of hills and valleys, lots of green.  In the end it turned out he wasn’t really an alien, but a human from the far future. He was pretending to be an alien as he was from 20000 years into the future and humans' appearance had changed by then, but he had the ability to make himself look like current-day humans. When he was in his car/bed he reverted to his normal appearance, so he would hide under the blankets until he was able to assume current-day human appearance.  In the finale, it was revealed that he was never in any danger – the pursuers were testing him as part of a kind of coming-of-age thing. I remember a scene on a spaceship or something where his elders/parents where shown telling him that he was safe all along and that he "passed".  I also seem to remember them referring to "ancestors" and making sure they didn't find out, so the either the girl or boy could well have been a direct ancestor. How to start Ubuntu in Console mode Why can't I tame my ocelot? I'm trying to make a cat house for my zoo on Minecraft. I have some untamed ocelots and I also want a few tamed ocelots. Fish only puts them in breeding mode.