How to evaluate the following limit $$\displaystyle\lim_{x\to 0} \dfrac{e-\left(1 + x\right)^{1/x}}{\tan x}$$ I have tried to solve it using L-Hospital's Rule, but it creates utter mess. Thanks for your generous help in advance.
find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
How to reconcile the cashless human/Federation society with human traders and merchants Getting "Authentication is required" when trying to access the Play Store Over the past week, whenever I try  to open the Play Store/Market app on my phone, the app thinks for a while and then displays this error message:     Authentication is required. You need to sign into your Google Account.   But I am indeed signed into my Google account, as seen in other Google applications running on the phone, and as seen if I open the menu in the Market app and click the "Accounts" button.  I suspect this might be related to the fact that over the past few months I've had two different Google accounts synced to this device, and I have recently removed one of them from the phone. No other Google application has had any problems with removing one account.  Does anyone know how I can solve this issue? I can still install applications by using the market website (now "Google Play"), but this is still a big inconvenience. I also think I stopped getting updates. I want the QED symbol to be 1 line lower at the end of proof How do question bookmarks work? What does bookmarking a question mean? How do you bookmark a question? Do users get updates on their bookmarked questions? Are my bookmarks publicly visible? How do you see your own bookmarks? How do you see other users' bookmarks? How do you see which users have bookmarked a particular question? What impact does the bookmark system have on the voting system? Can I &quot;un-bookmark&quot; a question? Can I search within my bookmarks? Listing multiple compound words How do I list multiple things that are compound words?  In my own language I'm used to writing these lists like this:  "I kicked foot-, basket-, and volleyball"  The sentence is supposed to mean that I kicked all types of balls (football, basketball and volleyball). However, it does not seem correct to me. Should I write them as whole words, with or without hyphen, or how? How do you run update-grub? I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. In Minecraft, there are a lot of cool sounds. I decided to find there location and move them to my desktop. I have Ubuntu, and if when I remove them from that folder will Minecraft be unable to play them or is it smart enough to figure out that they are gone and download them again. Before minecraft was installed, it had to download the files. I even moved my favorite song (creative 6) from the sounds. The sounds are not labeled and have seemingly random names like b0b12118a97ddd733a50e9382e91a21652568641 and how will I know what to call them that will be recognized without digging through the index if Minecraft fails to download them? Or do I not have to worry because it will download them again upon launch?  Note: Why does it not label them and call them random names? Model Selection: Logistic Regression In how many different ways can you split $10$ people into two groups with the same amount of people?  My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the combinations: $10 \choose 5$ What do we mean by hyperparameters? Can anyone give me full details about what we mean by hyperparameters, and what in the Dirichlet distribution are called hyperparameters? A practice example for the estimation of those parameters would also be useful. How to make a "reflective tape" material in cycles? How can I make a cycles material that gets very bright when a light is shown on it, yet still has normal colors? I'm talking about the "" that can be seen on road signs, the tape on big trucks (like in the image below), traffic barricades and clothing.  Image by TollGroup on    This reflective stuff has a interesting property where the colors show up perfectly well, but when exposed to a light source they seem to reflect much more light the normal.  has lots of good reference images.  I have tried different combinations of glossy and diffuse shaders, but none work as I have yet to get the super bright reflections.  How can cycles make a material that reflects so much light, and keeps the colors correct? Is using "fruits" as the plural of "fruit" acceptable? How do I install a different Python version using apt-get? How can I install a different version of Python using apt-get?  Obviously I realise I can install using the source tar ball, however I would prefer not to install from source and instead use the package manager, as that's what it's there for. Surely somewhere reputable builds .deb files for the latest Python releases (why python.org don't is beyond me) that I can reference.   What do I need to do to reference them and what issues might it create when upgrading to the next version?  If there is no way except for building from source, is there a (pseudo) package that I can can install that will provide all of the dependencies needed without having to find and install each individually? So that I don't get:  The necessary bits to build these optional modules were not found: _bz2                  _curses               _curses_panel       _dbm                  _gdbm                 _lzma               _sqlite3              _ssl                  _tkinter            readline              zlib I have a conjecture that any two positive coprime integers $a$ and $b$ can be added to form any integer greater than or equal to $(a-1)(b-1)$.   This is true for $3$ and $5$, which can form any integer greater than or equal to $8$, where $8=3+5, 9=3+3+3, 10=5+5; k=3n+8$ or $3n+9$ or $3n+10$.  It is true for $4$ and $7$, which can form any integer greater than or equal to $18$, where $18=4+7+7, 19=4+4+4+7, 20=4+4+4+4+4, 21=7+7+7$, etc. How can it be proven that this will work for any two coprime numbers? If the second field of the /etc/shadow file is set to !!, does it mean that the account is disabled? Or does it mean that the account is active with no password assigned? $\zeta_{15}$ is $15$th primitive $n$th root of unity.   Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$  I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})=\mathbb{Z_{p}}^*$ but when $p$ is not prime, I am not show how to solve this if not prime.  Does my required group have any relation to some cyclic group $\mathbb{Z_n}$, i.e. abelian  Would really appreciate your guidance as I have not got my head around these concepts of cyclotomic fields and Galois structure. Thanks Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. No matter how I do it, I always end up with $x = 0, 90, 270$ and $360$. All of those except $270$ is right, but I can't quite figure out how to get the $270$ degrees out of the answer. I've tried using trig identities, I've tried squaring both sides, but I always end up with $$2\sin x\cos x$$ which then leads me to $x = 0, 90, 270, 360$. But $$\sin (270) + \cos (270) = -1$$ so I'm doing something wrong. I am using QGIS 3.2.0 on a Mac (and also have QGIS 2), and I have a fishnet grid (probably created in Arc) that comprises of nearly 2 million grid cells. As a result, every change I make to the attribute table takes hours, so I can't experiment as I would usually.  I need to label each grid cell such that the north-westernmost cell is "1", and the numbers proceed by rows, until the last number goes into the south-easternmost cell. Using the row-number doesn't work, as that's not how the table is sorted. I've done this before, but cannot remember how.