How may I simplify the following so I could calculate the limit without getting $0$ in the denominator when replacing $x$ with $1$? $$\lim_{x \to 1} \frac{\sqrt{x}-x^2}{1-\sqrt{x}}$$
$$\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$$  I'm not entirely sure how to go about this, since rationalizing anything doesn't quite work for me. Just looking for some help.
reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time? iOS Run Code Once a Day Is it possible to prevent LUDecomposition from pivoting? How to change file encoding in NetBeans? How to print the full NumPy array, without truncation? Book Recommendation for mathematical finance Does anyone know a book which covers topics on:  Brownian Motion   Martingales   Stochastic Calculus  Stochastic Differential Equations   Options pricing. Black-Scholes model  Fundamental Theorems. Interest Rates   Random Walk   Applications in Insurance   Simulations. Convergence   Simulation methods   I would like something in-depth, but at an undergraduate level. How were bullets able to dent Captain America's Shield? Just after Captain America's shield was introduced in Captain America: The First Avenger, Peggy Carter shot few bullets at it which created few minor distortions on it.                                  From  page of Marvel.Wikia,     Proto-Adamantium   This is the original Adamantium created by Dr. MacLain. Proto-Adamantium is the official name for the metal that makes up Captain America's Shield. It has never been recreated and is the absolute most invulnerable item in all of existence.    From  page of Marvel.Wikia:     The true shield was once slightly dented by an enraged Odinforce empowered Thor, using Mjolnir, but he later reconciled and pounded out the small dent.    How were few bullets able to dent Captain America's shield then? Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ I was generalizing the following integral: $$\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x \hspace{40pt} n\geq 0$$ We can start by noting that $\sin x =\dfrac{e^{ix}-e^{-ix}}{2i}$ and thus $\displaystyle \sin^{2n+1}x=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\sin(2r+1)x$. This implies $$\begin{aligned}\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x &amp;=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\int_{0}^{\infty}\frac{\sin(2r+1)x}{x}\mathrm{d}x \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n(-1)^r\binom{2n+1}{r} \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n\left((-1)^r\binom{2n}{r}-(-1)^{r-1}\binom{2n}{r-1}\right)\end{aligned}$$ Where I have used the well known  $\displaystyle \int_{0}^{\infty} \frac{\sin(2r+1)x}{x}\mathrm{d}x=\int_{0}^{\infty}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$ and the property of binomial coefficients that $\displaystyle \binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$ Since the above sum telescopes, we have $$\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x=\frac{(-1)^n\pi}{2^{2n+1}}(-1)^n\binom{2n}{n}=\frac{\pi}{2^{2n+1}}\binom{2n}{n} ~\forall ~ n\in \mathbb{Z^{+}}$$ I would like to know other methods for evaluating this integral. Selecting the Real Analysis Textbooks I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on this Fall) on the analytic number theory, and prepare for the real analysis course I will take on Fall and Putnam competition. I just finished "Calculus with Analytic Geometry" by G. Simmons, "How to Prove It" by Daniel Velleman, and "How to Solve It" by G. Polya. I also read some portions of Apostol's Calculu Vol.I to get a deeper view on the calculus theories. I was originally planned to read Apostol's Calculus Vol.I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all ideas in those "advanced calculus" textbooks and much more. My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's PMA (required textbook for my real analysis course) starting on Summer and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category?   Elementary Real Analysis textbooks:  **Elementary Analysis: The Theory of Calculus (Kenneth Ross)  **Understanding Analysis (Steven Abbott)  **The Way of Analysis (Robert Strichartz)  **Real Mathematical Analysis (Charles Pugh)   Intermediate, detailed Real Analysis textbooks:  **Mathematical Analysis (Tom Apostol)  **Undergraduate Analysis (Serge Lang)  **Introduction to Real Analysis (Bartle, Sherbert) **Elements of Real Analysis (Bartle, Sherbert)  **Mathematical Analysis I (Vladimir Zorich)   Thank you very much for your time, and I look forward to your advice!   Sincerely, PK With left nav, responsive design, and various new-user notices, this is what an anonymous user currently sees (browser window here is 1090px wide):    I realize that a lot of this content -- code of conduct, new-user intro, cookies notice -- is important.  And I get that the left nav is an important design element, and responsive design dictates the width of the right column.  But when you add it all up, Q&amp;A -- what the site is about -- gets a distressingly small portion of the screen.  Can we do better?   Maybe we can make that intro/mini-tour shorter? Maybe we can hide the right column for anonymous users? Maybe some things can wait until the user's viewed a few pages? Maybe other things I haven't thought of?   My concern is that anonymous users -- people coming from Google, people who were referred to a site by users hoping to recruit them, or others who find their way to us but aren't invested in us -- will turn away before seeing how rich and helpful our sites are.  Which would be a real shame, because our sites are awesome.  First impressions matter.  When I'm asking a rabbi to check out Mi Yodeya or a professional chef to check out Seasoned Advice or my vet to check out Pets (all of these have happened), I want to make a good impression.  I want the person to see more than 2.5 questions; I want to increase the chances the person will see something interesting or say "oh I can answer that!" and dig deeper.  Yes, the other stuff is important, but maybe not now?  Is there a better way to "stage" all this information for people we haven't yet hooked? Product of manifolds & orientability I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable.   I am able to prove that the product is orientable if components are orientable (chart is $\{(U_\alpha\times V_{\beta},\phi_\alpha\times \psi_\beta):(\alpha,\beta)\in A\times B \}$, and $\det J=\det J_1 \det J_2&gt;0$ by Cauchy-Binet's theorem), but I don't know how to prove the other direction.   So why this holds: if $M\times N$ is orientable, then $M$ and $N$ are orientable?  Thanks in advance. Foo* set = new Foo[100]; // ... delete [] set;   You don't pass the array's boundaries to delete[]. But where is that information stored? Is it standardised? Correct user names when tracking /etc/ in git repository and committing as root We use git to track changes in /etc/ on our servers.  Administrators work as root when changing files in /etc/, and thus their commits have author  root &lt;root@machinename&gt;   This is not very satisfying since you cannot see which admin actually did the change.  What can we do to get the real admin names in the git log? I don't think that keeping a local clone of the repository is feasible since we often change interatively until something works, and a change-commit-push-seeError-repeat cycle would not help here. There is a statement that maximizing the likelihood is equivalent to minimizing the cross-entropy.  Are there any proof for this statement? How to execute multiple statements in a MATLAB anonymous function? I'd like to do something like this:  &gt;&gt; foo = @() functionCall1() functionCall2()   So that when I said:  &gt;&gt; foo()   It would execute functionCall1() and then execute functionCall2().  (I feel that I need something like the )  EDIT:  functionCall1 and functionCall2 are not necessarily functions that return values. Changing deleted partition back to NTFS without having to recover to another drive Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. I just moved to a new computer: Windows 10 with XAMPP. As far as I know I set everything up the same as it was on my old Windows 7 computer. But now, for some reason, much of the time when I run drush cc all or run update.php, the database gets corrupted. Most frequently, some cache tables' *.idb files (tablespaces) are deleted from my mysql directory, which makes those tables appear as "in use" in phpMyAdmin and prevents them from being edited or emptied. They are always cache tables and never tables that actually get modified by my update hooks. Restoring the database is the only way to fix it then. This is a mature site that I've been working on for a long time, and have never seen this until the new computer. What's happening here?  A sample error message when using update.php, caused by a missing *.idb file:  PDOException: SQLSTATE[HY000]: General error: 1812 Tablespace is missing for table '(null)': DELETE FROM {cache_field} WHERE (cid = :db_condition_placeholder_0) ; Array ( [:db_condition_placeholder_0] =&gt; field:message_type:6 ) in cache_clear_all() (line 168 of C:\&lt;path&gt;\includes\cache.inc).   My most recent drush cc all error message, which Google has never even heard of, was:  exception 'PDOException' with message 'SQLSTATE[HY000]: General      [error] error: 1030 Got error -1 "Internal error &lt; 0 (Not system error)" from storage engine InnoDB' Fatal error: Call to undefined function add_action() - an untouched problem The question was asked many times in many ways, but unfortunately is not covered well in most of the answers. I'm having the same issue doing things using include-files.  In my TwentyFourteen child theme's functions.php I used:  require_once get_stylesheet_directory_uri() .'/admin-panel/theme-options.php';   And I'm using . I've used it in many of my projects, the same way mentioned here, without having the error. But recently I'm having the issue in several of my projects. If I take the code to functions.php then everything's fine. But it's occurring only when I'm trying to include/require external file.  I'm having the error:     Fatal error: Call to undefined function add_action() in C:\xampp\htdocs\project\wp-content\themes\my-theme\admin-panel\theme-options.php on line 2   In this case I'm having the error using a child theme (of T14). But even in one of my parent theme I'm facing the issue. But in the same environment my other projects are running well with all the includes and/or requires (where several add_action()s are defined). I'm having issues in ALL the add_action() calls, like CPT assigning, Custom Taxonomy assigning etc. But I'm successfully developing another two projects including external files from inc/ folder where all my CPTs are assigned and their meta boxes too.  I tried reinstalling WordPress, deactivating all my plugins, tried from Child Theme, tried only the same thing by modifying TwentyFifteen (applying default theme), tried require(), require_once(), include(), include, require_once, and require one after another with failure. And I'm using WP 4.1.1 in all the good and problematic installs.  Majority of the previous questions are answered with a simple line:     As the message says, you haven defined the function add_action() and yet you're trying to use it.   But,   how come (isn't it already defined by WordPress)? and why it's differing project to project in the same environment (How come it's out of the scope now only in particular project[s])? To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). Why can't I travel onwards if my bag wasn't going to make the flight? I was flying KLM from the UK to the USA with a short layover in Amsterdam.  Because of delays in the UK, I almost missed my flight.  The flight attendant told me my bag wasn't going to make it, but if I walk fast, my gate is just two over.  When I got to the gate, they had already closed the doors, but the counter was opened - I mentioned that my bag was not going to make it, but that's okay.  She said, "hold on," got onto her walkie-talkie, and said,  "Yep, that's what I thought: you can't fly if your bag's not going."  I originally thought that it might be a security concern, but that doesn't really make any sense.  Why can't I travel if my luggage doesn't make the connection?  EDIT: I should clarify that I have no beef with KLM - they treated my overnight in Amsterdam professionally and helpfully; I'm just curious what the reason would be.