Prove or disprove: if $\int_1^\infty f(x)\,dx$ converges, then $\lim\limits_{x\to\infty}f(x) =0$.
Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?
Accessing private member variables from prototype-defined functions If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $ Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ What is IEEE 802.3 Ethernet's communication type: Synchronous or Asynchronous? In the IEEE 802.3 frame structure there is a 7 byte preamble section for synchronization. If it is really for synchronization does that mean Ethernet is synchronous and not asynchronous?    Ethernet structure for reference: I want to know, if this bug is fixed, because all previous versions of Ubuntu didn't work for me. So I just want to know, should i leave Windows 10 and go for Ubuntu 17.10. Thanks! How to tell the probability of failure if there were no failures? I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of the next 10,000 products sold fail? How to custom (split in two)? Ibidem When I login, nothing happens.   I am presented with my desktop wallpaper.    No Dash, no Launcher, nothing. The  mentions that it's possible to post one more question after a six-months waiting period and if the question is good enough, the question ban can be lifted:     If you're unable to improve your existing questions, you'll get the chance to ask one new one 6 months after your last question. If that question is positively received, you may be able to continue asking questions; if not, then the ban will be reinstated.   What is positively received in this context? Is it enough if the question has a zero score and is not closed? Or does it have to have a positive score?  How long does the system wait after posting this new question to evaluate this criteria and possibly lift (or reinstate) the question ban? Understanding how references and labels work I'm trying to understand why I cannot effectively reference figures or equations that I have defined with labels.  For example:  \begin{figure}   \label{fig:area_Treasuries}   \caption{Treasuries as a portion of total financial flows.}   \centering   \includegraphics[width=0.75\textwidth]{graphs/Area_Treasuries.png} \end{figure}  \paragraph{} Figure~\ref{fig:area_Treasuries} presents Treasuries (both flowing into and out of the United States) as a share of total financial flows.  From 1985 to 2010, Treasuries account for, on average, 53.2\% of total financial flows.   With the above code, I get this warning:      LaTeX Warning: Reference `fig:area_Treasuries' on page 37 undefined on   input line 477.   The result is "Figure ?? presents Treasuries..."  Why is this happening?  Related, I don't understand why I can't reference, say, an equation before (I mean physically above in the text) the actual equation.  Isn't this what compiling multiple times and cross references is supposed to take care of? I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? carrier concentration - distance graph of PN junction i am trying to plot the distance-carrier_concentration graph above with these formulas:   -and-  (p-side: x &lt; -xp , n-side: x &gt; xn) But i could draw only red lines on the graphs with these formulas. How can i or with what formula draw black lines on the graph that majority carrier and space charge region concentration? I am new and i need your help :) Thanks in advance. From Drupal 8 beta 6 (released today), I have this error :     Error message UnexpectedValueException: Untrusted Host "localhost" in   Symfony\Component\HttpFoundation\Request->getHost() (line 1221 of   core/vendor/symfony/http-foundation/Symfony/Component/HttpFoundation/Request.php).   I use this local url to call my site:   Inside my setting file, I have:       $settings['trusted_host_patterns'] = array(         'theming\.dev$',      );   Note, this error is displayed only when I uncomment this lines:   if (file_exists(__DIR__ . '/settings.local.php')) {    include __DIR__ . '/settings.local.php';  }   What is the problem please ? Thanks. How to read such figures with % as "up (by) 120%" and "increased (by) 120%". Do they mean the same? I'm too spoiled by Windows utilities that take a digital snapshot of your entire drive, which you can then restore from in the event of a drive crash.  (e.g. like Time Machine for Mac OS X).  Is there a similar way of doing this in Linux? What is it called when you use many words to describe the word you want to use? What is it called when you use many words to describe the word you want to use?  I know there is a technical word that's used in foreign language acquisition that means using many words to describe a word you don't know.  I'm doing it now, just talking and beating around the bush until someone can tell me the word I'm looking for. I have edited (created) the php.ini file and the .htaccess file in accordance with  post. I have also found a plug-in (called Upload Max File Size) that claims to raise the file size limit of media files. With the plug-in activated, I see on the add new media screen that the limit has been changed. However, when I attempt to upload a 2.7MB image, I receive the error:  The uploaded file exceeds the upload_max_filesize directive in php.ini.   The contents of php.ini are:  upload_max_filesize = 32M post_max_size = 32M max_execution_time = 300   My .htaccess file contains:  # BEGIN WordPress  php_value upload_max_filesize 100M php_value post_max_size 1M php_value max_execution_time 300 php_value max_input_time 300  # END WordPress   I am currently hosting the word press site myself and have root privileges to anything. I tried to install ffmpeg in trusty/Ubuntu 14.04 and got the following message:  $sudo apt-get install ffmpeg  Reading package lists... Done Building dependency tree        Reading state information... Done Package ffmpeg is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'ffmpeg' has no installation candidate   Why isn't the package available? Git SSH error: "Connect to host: Bad file number" How can I define a mathematical function as a LaTeX macro?