How can I calculate this limit?   $$\lim_{x\rightarrow 0} \frac{(1+x)^{\frac{1}{x}}-e}{x}$$     I thought about L'Hospital because case of $\frac{0}{0}$, but I don't know how to contiune from this point..
I am stuck on how to calculate the following limit: $$\lim_{x\to\infty}x\left(\left(1 + \frac{1}{x}\right)^x - e\right).$$  Definitely, it has to be through l'Hôpital's rule. We know that $\lim_{x\to\infty} (1+(1/x))^x = e$. So, I wrote the above expression as $$\lim_{x\to\infty}\frac{\left(1 + \frac{1}{x}\right)^x - e}{1/x}.$$Both numerator and denominator tend to zero as x tends to infinity. I applied l'Hôpital's rule twice, but I got the limit equal to infinity which is clearly wrong. In the book, it says that the limit should be $-e/2$.  Any help please? Also, in case someone managed to solve it, please tell me which numerator and denominator did you apply your l'Hôpital's rule to in both cases? Thanks a lot  Also, guys can you tell me how to write equations in this forum? Thanks.
How long does mana from Burning-Tree Emissary stay around? Why can't I catch Pokemon in Pokemon Go anymore? Does the selection of sRGB or Adobe RGB in camera when shooting RAW ever matter? I shoot RAW essentially 100% of the time, and have the in-camera color space setting at AdobeRGB (sRGB is also offered). The raw converter in turn is set to produce sRGB output files. As I understand the process, the general flow is for raw shooting: raw sensor data readout (including A/D conversion), compression, write to a raw file, then in post-processing rather than in-camera apply color space/color temperature/etc; for JPEG shooting: raw sensor data readout, apply color space/color temperature/etc, JPEG compression, write to memory card (in camera).  Hence, when a colleague said that I should leave the camera set to sRGB for best/consistent results despite using the camera's raw format, it didn't really make much sense to me. I have mulled it over for a while now and it still doesn't really make any sense.   seems to support my view of the differences in the data recorded in RAW and JPEG modes, respectively.  I did find , but the answers there seem to focus more on the difference between AdobeRGB and sRGB. My question is, when shooting RAW, does it ever matter (except for the image preview) whether the camera is set to sRGB or AdobeRGB? If it does, then why, and in what situations? Is my understanding of the "raw" format processing chain flawed?  In case it matters, I have a Canon EOS 50D. Travel to Australia with new passport (with visa in old passport) I was granted a visa to Australia few months ago. I’d like to travel to Australia in few days. However, I’ve just renewed my passport and I don’t have enough time to inform them about my passport renewal. Can I travel to Australia by carrying both my old and my new passports? I have two tables in my database:  NEWS table with columns:   id - the news id  user - the user id of the author)   USERS table with columns:   id - the user id   I want to execute this SQL:   SELECT * FROM news JOIN users ON news.user = user.id    When I get the results in PHP I would like to get associative array and get column names by $row['column-name']. How do I get the news ID and the user ID, having the same column name? Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$ How do you switch from Wayland back to Xorg in Ubuntu 17.10? Some of my applications don't work on Ubuntu 17.10 Wayland. How can I switch back to Xorg? I saw a thing called VPN connections under the Internet button at the top of my screen.   So I would like more info on this, and possibly instructions on how to set one up. Thanks! Proving $\frac{x}{y} +\frac{y}{z} + \frac{z}{x} \ge 3$ for positive $x,y,z$ Solve $x\equiv 1\pmod2$, $x\equiv 2\pmod3$, $x\equiv 3\pmod4$, $x\equiv 4\pmod5$, $x\equiv 5\pmod6$ and $x\equiv 0\pmod7$ $$\begin{align*} x&amp;\equiv 1\pmod2\\ x&amp;\equiv 2\pmod3\\ x&amp;\equiv 3\pmod4\\ x&amp; \equiv 4\pmod5\\ x&amp;\equiv 5\pmod6\\ x&amp;\equiv 0\pmod7\\ \end{align*}$$  So the solution says we can eliminate $x\equiv 5(\bmod6)$ because the first two cases cover it, but I don't really know how it does. How do we solve it in cases like this where the moduli are not mutually relatively prime. Why does Ubuntu 14.04 LTS use a non-LTS kernel version? How can I search tags on StackOverflow Careers? I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). No Wifi-Adapter found on new Lenovo Yoga C930 with Ubuntu 18.10 Complete classification of the groups for which converse of Lagrange's Theorem holds I am using natbib and agsm for harvard-style referencing. My related code is  \usepackage{natbib} \bibliographystyle{agsm}   Then I am using \citep{} for citations. My problem is that in text I am getting     (Author et al. 2012)   but and I want     (Author et al. 2012)   Does anyone know how to change this and get 'et al' in italics in the text?  I am using TexPad and BibDesk and I have not been able to find the .bst files in Terminal.  I know that it is often correct to have the 'et al.' unitalicised but my university regulations requires italics. I have been trying to install the latest version of java-8. However, I found out that the  due to changes in the java licence.  How should I update my java to 8.0_201 or 8.0_211 now?  Thanks! How do I make my document use the Times font, both for the text and the math? I know that I can put \usepackage{times} in the preamble in order to use the Times font instead of Computer Modern for my document.  But this only affects the text; the math is still typeset using Computer Modern.  How can I make Latex use Times also for the math? I am working on a git repository with a master branch and another topic branch. I have switched to topic branch and modified a file. Now, if I switched to master branch that same file is shown as modified.   For example:  git status in git-build branch:  # On branch git-build # Changes to be committed: #   (use "git reset HEAD &lt;file&gt;..." to unstage) # #       modified:   cvsup_current #   Switch to master branch  [root@redbull builder_scripts (git-build)]# git co master M       builder_scripts/cvsup_current Switched to branch "master"   git status in master branch  [root@redbull builder_scripts (master)]# git status # On branch master # Changes to be committed: #   (use "git reset HEAD &lt;file&gt;..." to unstage) # #       modified:   cvsup_current #   Why is that the file is shown as modified in master branch even though it was modified in git-build branch?   My understanding was that the branches are independent of each other and when I change from one branch to another the changes do not "spill over" from one branch to another. So I am obviously missing some thing here.   Anyone got a clue stick? What home printer would allow for exact alignment of front and back? Note: I'm not a graphic designer, but this seemed like the most relevant place to post. Feel free to send me off somewhere else if I'm not in the right place.  I would like to make some notebooks with double-sided grid paper that I've already designed, but my printer seems like it doesn't allow for absolutely exact alignment. Even if one side prints nice and straight, the other side will be slightly skewed left to right by one to two millimeters, which I read from an earlier post is a pretty reasonable variance for a home printer. So I suspect it's not possible to get both sides completely aligned on this printer – but if I'm wrong, please let me know how.  What kind of printer would allow for absolutely exact alignment? Is there any difference between inkjet and laser in this regard, or a more expensive printer versus a cheaper one? What can $E=mc^2$ do? In the famous equation , the variables stand for:     $E$ is energy, $m$ is mass, and $c$ is the speed of light (in vacuum).   And I understand the equation fairly but limited in knowing in its effects/outputs in reality, honestly.   Some example? Could anybody please give me some sort of examples in real world which are done by this equation?