Finding $\displaystyle \lim_{x\rightarrow 0}\frac{(1+x)^{\frac{1}{x}}-e+\frac{ex}{2}}{x^2}$ without using series expansion.   I am trying to solve it  using D L Hopital Rule  So $\displaystyle \lim_{x\rightarrow 0}\frac{(1+x)^{\frac{1}{x}}\cdot \frac{1}{x^2}\bigg[\frac{x}{1+x}-\ln(1+x)\bigg]+\frac{e}{2}}{2x}$  again i want to Diff with r to $x,$ but this is to complex  please help me how to short my calculation , Thanks
$$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$   by directly substituting $x=0$ i got $\infty$ by using L-H's rule, i got $-1/8$   the given options are  $a)\frac{24e}{11}$    $b)\frac{11e}{24}$    $c)\frac{e}{11}$    $d)\frac{e}{24}$    may be the question would be  $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$
I am having issues with input in VMware Workstation Player 12 in windowed mode or on multiple screens. When typing something in the VM and moving mouse cursor out of the VM without clicking on anything (for example on the second screen with host OS), the typing in that VM stops until I move the cursor back into VM.  Can this feature be disabled somehow so I am able to type in the virtual machine even when cursor is outside of it?  Both guest and host are Win10 x64 OS. I want to run a daily incremental backup and append the day of the week to the file name so I end up with:  backup_mon.bak backup_tue.bak  etc.  Not really bothered about what exactly is appended - 0 - 6 is fine.  How do I go about this? How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?  I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.  (the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution) How to change the style of the title attribute inside an anchor tag? In the Prisoner of Azkaban, Harry and Lupin could practice the patronus on the boggart, but when Harry was in the final task in the Goblet of Fire, he tried to use the patronus on the boggart, it didn't work and he then moved on to use ridikkilus.  Why could he use a patronus against the boggart, but then not in another? CASE WHEN with GROUP BY I have a table that looks like this (oversimplified case of the actual case to make my point)  CREATE TABLE sales (    id int,    currency VARCHAR(5),    price int    shop_id int );   Now I need to make a query that will give me something like this table, grouping by shops yet giving sum of sales per currency My problem is when I make currency column non aggregate it fails because my query becomes non standard. When I add currency to group by then each currency creates new row, something I don't want. I have run out of option and would appreciate any help or pointer to the right direction.  I use MySQL but would appreciate if I would get the solution that works with more databases (PostGreSQL being at the top of the list after MySQL)  Query that works with undesired results  SELECT   shop_id,           (CASE WHEN currency= "GBP" THEN SUM(price) ELSE 0 END) AS POUND,           (CASE WHEN currency= "USD" THEN SUM(price) ELSE 0 END) AS USDOLLAR GROUP BY shop_id, currency   The data that should be produced by the query  shop_id     |   POUND   |   USDOLLAR     -----------------------------------------     1       |   400     |   300 -----------------------------------------     2       |   250     |   100 -----------------------------------------     3       |   400     |   100 ----------------------------------------- Printing a file to a printer in Ruby Basic Excel currency format with Apache POI I'm able to get cells to format as Dates, but I've been unable to get cells to format as currency... Anyone have an example of how to create a style to get this to work?  My code below show the styles I'm creating... the styleDateFormat works like a champ while styleCurrencyFormat has no affect on the cell.  private HSSFWorkbook wb; private HSSFCellStyle styleDateFormat = null; private HSSFCellStyle styleCurrencyFormat = null;   ......  public CouponicsReportBean(){     wb = new HSSFWorkbook();     InitializeFonts();  }  public void InitializeFonts() {     styleDateFormat = wb.createCellStyle();     styleDateFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("m/d/yy"));       styleCurrencyFormat = wb.createCellStyle();     styleCurrencyFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("$#,##0.00"));  } What's the best solution to check a users country? I was looking for proofs using Calculus for the area of a circle and come across this one $$\int 2 \pi r \, dr = 2\pi \frac {r^2}{2} = \pi r^2$$  and it struck me as being particularly easy. The only other proof I've seen was by a teacher and it involved integrating $x = \sqrt{r^2 - y^2}$ from $-1$ to $1$, using trig substitutions and then doubling the area to get $\pi r^2$ but the above proof seemed much more straight forward.  Is it a valid proof, or is it based on circular logic or some other kind of fallacy? How do I make Panel always transparent like the Dock in GNOME 3? I'm using Ubuntu 17.10. I want the Panel to be always transparent instead of dynamic transparency. Subfields of finite fields I'm trying to identify a science fiction series on UK TV during the 70s, but I can only recall the following:   A party of adults and children. There was a woman with a telepathic bond to her cat: she could look into its eyes to see what it had seen. Time travel was involved. I think it may have been inadvertent or uncontrolled time travel. On one occasion the party encountered an Elizabethan seaman who demanded to know if Elizabeth was still queen. They party replied "err... yes" because they didn't want to have to explain time travel. Why can't I press the Install button when installing applications from unknown sources? I'm trying to sideload an application, as seen in the picture below.    I have allowed the installation of applications from unknown applied actions in settings, but I can't seem to press the install button. The cancel button works correctly and installing applications via adb install works correctly. What's the problem?  EDIT: For those who have this problem, using ChainFire's  will correct it because the app uses hardware compositing, and Android doesn't notice a "layer" on top of the current screen. Is it possible for information (like 1 and 0s) to be transmitted faster than light?   For instance, take a rigid pole of several AU in length. Now say you have a person on each end, and one of them starts pulling and pushing on his/her end.   The person on the opposite end should receive the pushes and pulls instantaneously as no particle is making the full journey.  Would this actually work? IS or ARE? "The only thing that I want you to hit right now IS/ARE the books" Exclude menu item from the collapse of bootstrap 3 navbar How to position the paper's authors' addresses on the front page (maketitle)? When I use the following code for the title page of a paper (please see below), the authors' addresses and emails appear on the last page of the paper. However I need them to appear on the first-title page (after the authors' names before the abstract). How can I do this?  \documentclass [reqno, 12pt ]{amsart} ... \begin{document}  \title[paper]{Nice paper}  \author{Me} \address{University} \email{mmm@lala.com} \thanks{.} \author{my friend} \address{Institute} \email{kkk@lala.edu} \thanks{.}  \date{\today} \keywords{XXX }  \begin{abstract} Lalala \end{abstract}  \maketitle I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some constant, then that should be continuous, but if we formulated continuity in terms of images it won't be continuous. Is the reason we do it that way, because pre-images will always guarantee that function in the domain can only have $1$ value? Why is quadratic integer ring defined in that way? Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation}    \mathcal{O}=\begin{cases}       \mathbb{Z}[\sqrt{D}] &amp; \text{if}\ D=2,3\ \pmod 4\\       \mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]\  &amp; \text{if}\ D=1\pmod 4    \end{cases} \end{equation} where $D$ is square-free.   I understand $\mathbb{Z}\left[\frac{1+\sqrt{D}}{2}\right]$ is not closed under multiplication if $D=2,3\pmod 4$. But still, isn't it more natural to define $\mathcal{O}=\mathbb{Z}[\sqrt{D}]$ for all square-free $D$? (in that case, it really seems like 'quadratic integer') I wonder what is the motivation of this definition.