As the title states, I want to prove  $$\lim_{x \to \infty} \frac{x}{e^x} =0$$  Clearly, L'Hopital's rule easily solves this. However, I'm curious to see if there's another way to prove it, without involving some differential or integral calculus (that is, by algebraic means).  What I'm really interested about, is to prove that  $$\lim_{x \to \infty} \frac{x}{e^{x^2}}=0 $$ I assume that proving the first limit will provide a way to prove the second one, using the squeeze method. If you know a direct way to prove the second limit, it will be more than perfect.   Thanks in advance!
I want to prove that $xe^{-x}$ is bounded by some constant for all the $x \in [0, \infty)$. So I take derivative of this guy, and found that on $x \in [0, \infty)$, its derivative is positive from $[0, 1)$, $0$ at $x=1$, and negative from (1, in infinity). So  $xe^{-x}$ achieves its maximum in $[0, \infty)$ at $x=1$, which is $1/e$. However, to prove that it's bounded by $1/e$, I need to prove that as $x$ goes to infinity, $xe^{-x}$ converges to $0$. Because otherwise, if it went to some value below $(-1/e)$, $xe^{-x}$ would not be bounded by $1/e$. How can I prove that $xe^{-x}$ converges to $0$? I mean, as $x$ goes to infinity, $e^{-x}$ converges to zero, but at the same time, x diverges to infinity. How can I formally prove that the product term converges to $0$? Do I need to use the epsilon definition of convergence to prove it?  Could you just show me the formal proof of $xe^{-x}$ converges to $0$? Or is there's any other way to approach this question? My professor suggested that I should use the power series for $e^{-x}$...?   Your help is greatly appreciated! Thank you so much!  I hope you enjoy your Saturday!
How to select the architecture to download? Is accessing a variable in C# an atomic operation? I've been raised to believe that if multiple threads can access a variable, then all reads from and writes to that variable must be protected by synchronization code, such as a "lock" statement, because the processor might switch to another thread halfway through a write.  However, I was looking through System.Web.Security.Membership using Reflector and found code like this:  public static class Membership {     private static bool s_Initialized = false;     private static object s_lock = new object();     private static MembershipProvider s_Provider;      public static MembershipProvider Provider     {         get         {             Initialize();             return s_Provider;         }     }      private static void Initialize()     {         if (s_Initialized)             return;          lock(s_lock)         {             if (s_Initialized)                 return;              // Perform initialization...             s_Initialized = true;         }     } }   Why is the s_Initialized field read outside of the lock? Couldn't another thread be trying to write to it at the same time? Are reads and writes of variables atomic? Prove that$$\overbrace{\binom n 0 + \binom n 1 + \binom n 2 + \cdots + \binom{n}{\frac{n-1} 2}} + \overbrace{\binom n {\frac{n+1} 2} + \cdots + \binom n {n-1} + \binom n n} = 2^n. $$   Background:  This problem came from the following link:    and I didn't understand why that was true in which no one on the page has proved it to be so. and i was wondering what was the proof on that What is the method to make files in a USB Flash Drive undeletable? Does these method work with USB 3.0 interface?  (I have surveyed some Flash Drive manufacturers, and I only see undeletable files for USB 2.0 drives, and not 3.0 drives. One manufacturer claimed that USB 3.0 cannot do undeletable file, so I am curious whether the claim is correct...) How do you create product mock-ups? Is acceleration relative in relativity? Suppose a box A is moving relative to a Box B, then by time dilation equation if I take 1 sec passed for an observer in A then for an observer in B will be little longer. Now if I suppose that the box B is moving while A is stationary under the same condition, then by the time dilation equation time passed in B must be shorter than A. How is this issue resolved? In the twin paradox problem my book says that it's due to acceleration, but in my opinion acceleration is relative (please correct me if I'm wrong), but here it is not the case. How do I instantiate a string array object into a Spring.Net container? What does "upgrade to 18.04" mean? I'm using Ubuntu 16.04 LTS on an Asus K53E laptop with 8 GB RAM. I check for updates to the software daily. If there are no updates, it asks me if I want to upgrade? What does upgrade exactly mean? Will my personal files still be there? Will it wipe EVERYTHING and give me a completely new installation of Ubuntu? Or will it allow me to keep my files and applications, and merely upgrade the OS?  When I upgrade my Mac or Windows machines, my personal files remain available after the upgrade. Does Ubuntu do the same, or are there completely different "rules"? I am trying to install Wordpress on Lubuntu. I followed the instructions . Including PHP Apache and MySQL setups. Apache and MySQL appear to be running fine.  However, when I go to http://localhost/blog/, I get the following error:     Neither /etc/wordpress/config-localhost.php nor /etc/wordpress/config-localhost.php could be found.   Ensure one of them exists, is readable by the webserver and contains the right password/username. LIKE uses index, CHARINDEX does not? I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate.  The famous differential equation problem models a skydiver jumping out of a plane as  $$m \frac{dv}{dt} = -mg-cv$$ where m is the mass of the skydiver, g is acceleration due to gravity (9.81 m/s) and c is the coefficient of air resistance.  Here is my question:     Suppose the skydiver exits the stationary helicopter at an altitude of 2000m and opens the parachute at 500m. For this exercise, the parachute opens instantaneously. Suppose m= 70kgs, the coefficient of air resistance is c = 14 for a free falling body and c = 105 after the parachute opens. When does the skydiver open the parachute? When does the Skydiver hit the ground?   So my attempt at a solution kinda starts like this, but this is where I get stuck:  Let's assume that s(t) is the distance the body falls in time t from its initial point of release, then $v = \frac{ds}{dt}$ and $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$. Thus we have a second order differential equation: $m \frac{d^2s}{dt^2} = -mg-c\frac{ds}{dt}$. By simply manipulation, we get $m \frac{d^2s}{dt^2} + c\frac{ds}{dt} = mg$. I originally started thinking of finding the integrating factor, however, this seems more complicated and won't help produce the equation that I need.  Would anyone be able to help? Much thanks in advanced! I only read part of the second book in this series years ago, and am unsure of how many books were in it. It was about a teenage or young adult boy, who was part of the ruling class of this fantasy world. He was gay, and his lover was the heir to the throne. The antagonist of the first book - I haven't read it, but I got the gist from comments in the second - was the current queen, who tricked the boy and his lover into leaving the kingdom for remote outer regions, thus enabling her son to take the throne. She then married her own son to keep herself in power.  The second book had the boy and his lover slowly becoming enemies, as the lover began to corrupt the simple tribesmen they were sheltering with in his quest to regain his throne. He wanted to build an empire in the hinterland and use the army to attack his homeland, killing the queen in the process, and didn't care about the long-term effects his actions were having on the culture or environment of his new followers.   The lover defeats a tribesman by breaking his neck in unarmed combat, then later takes a young boy as his "willing catamite." He also leads raids on neighbouring tribes, establishes alliances with other neighbouring tribes and organises daring and successful raids on the eggs of dinosaurs living in the middle of a swamp. He enslaves the first tribe to turn against him, and gives their children to slavers from the empire in place of those of allied tribes, thus weakening his enemies while ensuring the loyalty of his allies.  The series involved dinosaurs living among humans, being both domesticated and wild, used as livestock, for hunting and as transport. The boy and his lover were much taller than the average humans who were forced to pay tribute to this pseudo-magical empire they came from; all the humans from the empire were much larger than their subjects. The tribesmen have a tradition about not touching the ground with bare feet, as it is an insult to nature, or their god. The boy and his lover both ignore this superstition, but the catamite also begins to ignore it, despite looking ashamed as he does so.  I have absolutely no idea when this series was written, what it is called or who wrote it. All I know is that I read it in 2005-06, though it probably wasn't new. I'm hoping the long description I've given will help someone work it out. Rails: Could not find bcrypt-ruby-3.0.1 in any of the sources I tried to find this question somewhere else on here, and found a couple that were similar to it -- but all the answers seemed pretty convoluted, and I think it was surprising to me because I just imagined that SQL would have a way easier way to do this.  Essentially, I'm using a GROUP BY to group two values on top of each other. The coding is simple and works well, where I get these results using WITH ROLLUP at the end:  Type    Person  Count Type A  Person 1    10 Type A  Person 2    91 Type A  Person 3    13 Type A  Person 4    10 Type A  Person 5    2 Type A  NULL        126 Type B  Person 6    16 Type B  Person 7    63 Type B  Person 8    15 Type B  Person 9    22 Type B  Person 10   16 Type B  Person 11   3 Type B  Person 12   20 Type B  Person 13   44 Type B  NULL        198 NULL    NULL        360   Is there an easy way to substitute something saying if "NULL" is in the "Person" column, then make NULL AS "Subtotal" and if "NULL" is in the "Type" column, NULL AS "Grand Total?"  And if that's weirdly complicated, I'd be totally fine with just calling all of them "Total."  Thanks! Filling triangle doesn't seem to work with PGF/TikZ I was wondering if anyone had any clue on why my triangles are not filled in red within the following piece of code:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc,3d}  \usepackage[active, pdftex, tightpage]{preview} \PreviewEnvironment[{[]}]{tikzpicture} \newcounter{number}  \begin{document} \begin{tikzpicture}[x  = {(-0.5cm,-0.5cm)},                     y  = {(0.9659cm,-0.25882cm)},                     z  = {(0cm,1cm)}] \begin{scope}[canvas is yx plane at z=0] \foreach \x / \y / \label in {1 / 0.4 / 1, 0.9 / 0.4 / 0, 1 / 0.5 / 1, 0.7 / 0.3 / 0, 0.8 / 0.4 / 0, 0.9 / 0.5 / 0, 1 / 0.6 / 1, 0.7 / 0.4 / 0, 0.4 / 0.4 / 0, 0.6 / 0.4 / 0, 0.5 / 0.4 / 0, 0.8 / 0.5 / 0, 0.9 / 0.6 / 0, 1 / 0.7 / 1, 0.7 / 0.5 / 0, 0.5 / 0.5 / 0, 0.4 / 0.5 / 0, 0.6 / 0.5 / 0, 0.8 / 0.6 / 0, 0.9 / 0.7 / 0, 1 / 0.8 / 1, 0.7 / 0.6 / 0, 0.6 / 0.6 / 0, 0.5 / 0.6 / 0} { \addtocounter{number}{1} \pgfmathparse{\value{number}}\let\k\pgfmathresult \draw (8 * \x, 8 * \y) node[inner sep=0, outer sep = -0.1pt,color=black,fill,circle] (node-\k) {}; };  \foreach \x / \y / \z / \trash in { 4/5/8/0, 9/11/16/0, 9/16/17/0, 11/10/18/0, 11/18/16/0, 10/8/15/0, 10/15/18/0, 8/5/12/0, 8/12/15/0, 5/2/6/0, 5/6/12/0, 2/1/3/0, 2/3/6/0, 17/16/24/0, 16/18/23/0, 16/23/24/0, 18/15/22/0, 18/22/23/0, 15/12/19/0, 15/19/22/0, 12/6/13/0, 12/13/19/0, 6/3/7/0} { \draw[fill=red]  (node-\x) --  (node-\y) --  (node-\z) -- cycle; } \end{scope} \end{tikzpicture} \end{document}) Does adverb placement affect meaning? He swam slowly to the island.  He slowly swam to the island.    Some experts say that there is a “slight difference” in meaning. Would you please tell me that difference? Ambiguity of the word 'any'? What are the consequences of a US citizen overstaying a Schengen visa? biblatex, biber, is it possible to \printbibliography between two years? With biblatex and Biber, is it possible to print all the bibliography (\printbibliography) from (let's say) 2003 to 2008 in one section? And then in another section print all the bibliography from 2009 to 2016? I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups.  Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then $\operatorname{Aut}(G\times H)\cong\operatorname{Aut}(G)\times \operatorname{Aut}(H)$.  I also know that if $G,H$ are abelian and simple and isomorphic, then $\operatorname{Aut}(G\times H)\cong \operatorname{GL}_2(\mathbf{Z}_p)$ where $p$ is the order of $G$.  What I don't know is the following: what if $G,H$ are simple, yet not abelian? Let us for now focus on a more restricted case where $G\cong H$, in particular, $G=H$. What can we say about the group $\operatorname{Aut}(G\times H)$? (I heard something about this group being related to what is called "wreath product", which I do not know...)  Thanks bunch in advance!