I will soon make a math exame where one can't use L'Hôpital's rule, integrals concept neither the formal limit definition. The most I can use is the derivative definition and the algebraic ways to solve limits.  My thought was:  Let $\displaystyle \lim_{x \to \infty} \frac{\log(x)}{x}=y$, so $\displaystyle \lim_{x \to \infty}\log(x^{\frac{1}{x}})=y$.  Then,  $$\displaystyle \lim_{x \to \infty}e^{\log(x^{\frac{1}{x}})}=e^y \Leftrightarrow$$   $$\displaystyle \lim_{x \to \infty}x^{\frac{1}{x}}=e^y $$  I ended with an indetermination.  How can I proof the $\displaystyle \lim_{x \to \infty}\frac{\log(x)}{x}=0$ with the restritions and as formal it can get? Thanks.
Find $\space\ \begin{align*} \lim_ {x \to+\infty} \left [ \frac{\log_{2}(x-1)}{x}\right]   \end{align*}$.   After some minutes around this limit I did it this way:  $\log_{2}(x-1)=y \Leftrightarrow 2^y=x-1$  So,$\space x=2^y+1$.  When $x \to +\infty$,$\space y \to +\infty$ also. By substitution:  $\begin{align*} \lim_ {y \to+\infty} \left [ \frac{\log_{2}(2^y+1-1)}{2^y+1}\right]=\lim_ {y \to+\infty} \left [ \frac{\log_{2}(2^y)}{2^y+1}\right]=\end{align*}$  $\begin{align*}\lim_ {y \to+\infty} \left [ \frac{y}{2^y+1}\right]=\lim_ {y \to+\infty} \left [ \frac{1}{\frac{2^y+1}{y}}  \right]=\lim_ {y \to+\infty} \left [ \frac{1}{\frac{2^y}{y}+\frac{1}{y}}\right]= \frac{1}{+\infty+0}=0  \end{align*}$  Is this correct?Are there any other easy way to find this limit?Thanks
Cannot log into Area 51 with OpenID I recently tried many times to log into Area 51 with my Stack Exchange OpenID. Each time, I got this:    Then, I went to my OpenID account, and I saw this:    ...what is going on??  I have chrome, and extensions that block 3rd party cookies... is that the problem? A standard proof for press needs crop marks and color bars as below    With the package crop, the crop marks can be easily added.  But how about the color bar ?  Is there a way or a package to make it ?  An example is   \documentclass[b5paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage[cross,a4,center,info,axes]{crop}  \title{Test} \author{User} \begin{document} \maketitle \section{Fox} The quick brown fox jumps over a lazy dog. \end{document} Since , how do I uninstall it from my desktop system?  (Please note: This question is not about Unity 7, which is still supported at the time of this writing and should remain installed if present.) Notification that ‘someone might be watching what you browse’ I got a notification on my iPhone. The notification reads     Someone might be watching what you browse. Follow 3 steps to encrypt your web traffic and change your IP address.   What does this mean? What does "65" mean? K-2SO beats up the troopers easily. Up in the sky, the bombers are preparing to attack. Just as they start the attack run, TIE fighters come in from the shield and swarm the rebel fleet.  JON VANDER (GOLD LEADER): Red Leader, this is Gold Leader, we’re starting our attack run on the  shield gate. Keep it tight and watch out for those towers. GOLD PILOT: More fighters. Form up, at 65.  What does &quot;65&quot; mean? How does Salesforce SSO call just-in-time handler methods I am having a hard time digging up some information on this as I believe it is still relatively new. If you implement SSO with Salesforce you have the ability to enable SAML. I have enabled SAML and have also enabled JIT with an automatically created class to support it.  Out of this newly generated class, there are two global methods that I am assuming are called upon SSO, but I do not know which one is called or how.   global User createUser(Id samlSsoProviderId, Id communityId, Id portalId,     String federationIdentifier, Map&lt;String, String&gt; attributes, String assertion) {     User u = new User();     handleJit(true, u, samlSsoProviderId, communityId, portalId,         federationIdentifier, attributes, assertion);     return u; }  global void updateUser(Id userId, Id samlSsoProviderId, Id communityId, Id portalId,     String federationIdentifier, Map&lt;String, String&gt; attributes, String assertion) {     User u = [SELECT Id, FirstName, ContactId FROM User WHERE Id=:userId];     handleJit(false, u, samlSsoProviderId, communityId, portalId,         federationIdentifier, attributes, assertion); }   Does Salesforce have some backend code that interprets the SAML coming over the SSO and then calls the methods appropriately?   The reason I am asking this is because I would like to cancel a user's SSO if they do not match x fields based off of the values coming across in the SAML in the SSO. If I did not want to update a user or create a user, would I be able to add in my own global class that could be called from the SSO?   Let me know if this is unclear / does not make sense.  Thanks What is a concise function that formats a (String) decimal into a currency format in Apex? When do "per day" spell-like abilities recover in 3rd edition? I would like to have a paper's bibliography entries inside its .tex file instead of outside in a .bib file. Is this possible with biblatex?  My reason for asking this is that I want to find the most convenient way to produce documents that uses biblatex from just one file to make collaboration on papers easier and to be able to handle submissions of papers to journals. How do I bind a WPF DataGrid to a variable number of columns? My WPF application generates sets of data which may have a different number of columns each time.  Included in the output is a description of each column that will be used to apply formatting.  A simplified version of the output might be something like:  class Data {     IList&lt;ColumnDescription&gt; ColumnDescriptions { get; set; }     string[][] Rows { get; set; } }   This class is set as the DataContext on a WPF DataGrid but I actually create the columns programmatically:  for (int i = 0; i &lt; data.ColumnDescriptions.Count; i++) {     dataGrid.Columns.Add(new DataGridTextColumn     {         Header = data.ColumnDescriptions[i].Name,         Binding = new Binding(string.Format("[{0}]", i))     }); }   Is there any way to replace this code with data bindings in the XAML file instead? I want to install Win Xp on my Ideapad Z 570 laptop but i always get blue screen with message with some drive related problem which asks me to run CHKDSK command. I tried with that but no luck. Is there any way i can install Win Xp with SP 2 ?  Edit: i searched across the sites and came to know there could be some Compatibility problems, and i could fix it by setting HDD mode to IDE; will it solve the problem? How do I make Windows 10's File Explorer open "This PC" by default? I know how to do that on Windows&nbsp;Vista and Windows&nbsp;7, and I know how to do that on Windows&nbsp;8 and Windows&nbsp;8.1, but neither of the ways work with Windows&nbsp;10; it always defaults into the "Home" view.  (I am basically reposting  for Windows&nbsp;10.) Inverse of sum of exponential random variables (mean and variance) Assume $X_1,X_2, \dots , X_n$ following the exponential distribution with mean $\theta &gt; 0 $ and the statistic: $$ T = \sum\limits_{i = 1}^n X_i $$  I know that the sum of exponential random variables follows the Gamma distribution, but I cannot infer anything about the inverse of the sum $\frac{1}{T}$. My guess would be that: $$ E\left[\frac{1}{T}\right] = \frac{1}{\theta} \quad \text{and} \quad V\left[ \frac{1}{T} \right] = \frac{1}{\theta^2} $$ but why? \colorbox height too long \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{xcolor} \usepackage{minted} \usepackage{parskip}  \definecolor{codegray}{gray}{0.9} \newcommand{\code}[1]{\colorbox{codegray}{\texttt{#1}}}  \begin{document}  \code{test code}\\ anotherline  \end{document}   This is working good on sharelatex.com and looks like I expect and want it:   but on my local project using this command gives me this:   any idea what could be causing this colorbox height going to the next line? Given a compact set $K$ and closed set $E,$ prove: $E\cap K\neq \emptyset $ Let $K\subseteq \mathbb{R}^n$ be a compact set and $E\subseteq \mathbb{R}^n$ a closed set, such that: $$\inf\{d(x,y):x\in K, y\in E\}=0$$ where $d(x,y)=\sqrt{\sum_j(x_j-y_j)^2}$ Prove: $E\cap K\neq \emptyset $  I tried to work with the definitions of compact and closed sets, and I understand that from what's given of the distance between elements in $K$ and $E$, that they are very close,  but couldn't really get further. Any help appreciated. Some icons are correct, and some are whack. Turns out it's not just the hot questions list, but the whole icon set. Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? I have two questions. First, which command lists files and directories, but lists directories first?  Second question: I want to copy a list of files into a single directory, but make the target directory the first filename in the command. Drupal site moved from development to live - IE won't load stylesheets anymore Before you ask ... I have looked everywhere for a solution, even drupal.org but nothing so far seems to work.  I developed a Drupal site (custom child theme for Zen) and on the development server everything looks fine in all browsers.  I did a test migration to another server before the launch. This move went well and everything looks fine in all browsers.  Then I moved the site to its live server where everything looks fine in all browsers except Internet Explorer which is not loading the stylesheets.  I know Drupal is known for generating many stylesheets and Internet Explorer will not load them if there are more than 31 (I believe) of them, but I did set the "Aggregate and compress CSS files" feature and clear the cache. This did not seem to help.  Could the answer be in one of the two following things?   The development site and backup site are both with one hosting provider. The live site is with a different hosting provider. The development site and backup site are in subfolders of two different portfolio sites ... ie www.portfoliosite.com/Newsite. The live site is a subdomain of the parent site ... ie www.newsite.parentsite.com.   I have run out of ideas so if anyone has a possible solution or direction to look, please let me know. Thanks I couldn't add defer tag in plugin javascripts. Google developer pagespeed test suggests me to add defer tag in contact form 7 javascripts.   This is how contact form 7 includes javascript in header.  add_action( 'wp_enqueue_scripts', 'wpcf7_enqueue_scripts' );  function wpcf7_enqueue_scripts() {     // jquery.form.js originally bundled with WordPress is out of date and deprecated     // so we need to deregister it and re-register the latest one     wp_deregister_script( 'jquery-form' );     wp_register_script( 'jquery-form', wpcf7_plugin_url( 'jquery.form.js' ),         array( 'jquery' ), '2.52', true );      $in_footer = true;     if ( 'header' === WPCF7_LOAD_JS )         $in_footer = false;      wp_enqueue_script( 'contact-form-7', wpcf7_plugin_url( 'scripts.js' ),         array( 'jquery', 'jquery-form' ), WPCF7_VERSION, $in_footer );      do_action( 'wpcf7_enqueue_scripts' ); }   Now how to add defer="defer" tag in the above code?