$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$  How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} f'(x)=0?$$  Note I tried to take $f(x)=\frac{1}{x}$ but it did not work  Thank you for any kind of help
Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$.
Horizontal spacing issue in \listoftheorems using amsthm + thmtools A 256GB SSD has Windows-10 and 16.04 installed.  BIOS not UEFI.  A charity I work with requires me to use Quickbooks Desktop on Windows.  The plan is to ghost the existing system to a 512GB or larger SSD and create a third bootable Windows 7  partition for performing charity activities.   Is this possible to have a third partition? Is there a good example you have used as a guide? Is triple booting with two Windows partitions different from 2 Linux partitions?   If you have done this before ( with GRUB) please indicate this in your response: lessons-learned are appreciated.  Thank you Why Did Anakin Skywalker Believe that Turning to the Dark Side of the Force Would Save Padmé? Just as the question asks: Why did Anakin come to believe that Padmé could be saved if he embraced the Dark side of the Force? Padmé clearly didn't believe it. Why was Anakin so driven to save Padmé through the dark side? There are a lot of software in Windows to merge PDF files but how can we do the same in Ubuntu? What is the earliest work considered to be Science Fiction? Is it ok to put old papers on the arXiv? In my field (I am a condensed-matter theoretical physicist) it is such a normal thing to put papers on the arXiv, that people sometimes look at other people's arXiv list of papers instead of their publication record to get to know what they do, etc.  It just so happens that a few years ago I did a PhD with a supervisor that does not like the arXiv that much (he apparently had some conflict with some authors of a paper that was very similar to one of his papers, but one was already published while the other was just an arXiv preprint -or something like that). As a result, from my list of less than 20 publications, there are four of them from my PhD that are missing on the arXiv.  I was thinking of submitting the latest versions I have of them as arXiv preprints now, four years after their publication. For this I already have the agreement of all co-authors (including my supervisor), and all the journals involved seem to allow to put preprints of the material they publish. Do you see any problem with this? I fear that some people might take it as a way to advertise as new something that was already published a long time ago (though I would of course give the full reference to the published version). There may be other issues I am not aware of. The reason why I would like to do this is that I am applying for fellowships, positions, etc. and I am afraid some of my (best) publications may go unnoticed if somebody looks me up on the arXiv.  Thank you, Stephen Newtonian mechanics: massless particles Explaining two-tailed tests I am looking for various ways of explaining to my students (in an elementary statistics course) what is a two tailed test, and how its P value is calculated.  How do you explain to your students the two- vs one- tailed test? Adding DC voltages of PSU I need to add two voltages together that share a common ground (power supply from computer). I tried using bridge rectifiers and Op-Amps. How to add a new kernel to grub2 How can we add a kernel to grub2? i knew the stuff we could do with the former versions, but it seems to have changed. What files should we edit? Is there software on Macintosh for adding GPS data to RAW files? How does TikZ sort intersections? I used the sort by key for intersections to sort along a line, expecting the intersections to be numbered 1 through 5. Why were they numbered 4, 5, 3, 1, 2? Playing around with it, I could not seem to figure out how the order is determined. The code and image follow.  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{intersections}  \begin{document} \begin{tikzpicture}     \draw[name path=grid] [xstep=3,ystep=2] (0,0) grid (9,8);     \draw[-&gt;, name path=line] (2,1) -- (7,7);     \draw[name intersections={of=grid and line, sort by=line, name=i, total=\t}]         \foreach \s in {1,...,\t}{(i-\s) node {\s}}; \end{tikzpicture} \end{document} What does $\bigcup_{i=0}^{\infty}[i, i+2]$ mean? I think I get it but I wanted to be sure before attempting the question:  Question  Describe the following set, and prove your answers correct. (Here brackets denote intervals on $\mathbb{R}$.)  $\bigcup_{i=0}^{\infty}[i, i+2]$  Does it mean: {$0, 1, 2, 3, 4, 5...,\infty$} $\cup$ {$2, 3, 4, 5, 6, 7,...,\infty$}?  If so, what is the difference between:  $\bigcup_{i=0}^{\infty}[i, i+2]$ and $\bigcup_{i=0}^{\infty}(i, i+2)$ I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: Why is it better to use "#!/usr/bin/env NAME" instead of "#!/path/to/NAME" as my shebang? I notice that some scripts which I have acquired from others have the shebang #!/path/to/NAME while others (using the same tool, NAME) have the shebang #!/usr/bin/env NAME.  Both seem to work properly.  In tutorials (on Python, for example), there seems to be a suggestion that the latter shebang is better.  But, I don't quite understand why this is so.  I realize that, in order to use the latter shebang, NAME must be in the PATH whereas the first shebang does not have this restriction.  Also, it appears (to me) that the first would be the better shebang, since it specifies precisely where NAME is located.  So, in this case, if there are multiple versions of NAME (e.g., /usr/bin/NAME, /usr/local/bin/NAME), the first case specifies which to use.  My question is why is the first shebang preferred to the second one? What does the minus sign in Maxwell's third equation imply? If we write out Maxwell's equations with magnetic charges, we get $$ \begin{align} \nabla \cdot \mathbf{E} &amp;= 4 \pi \rho_e \tag{1}\\ \nabla \cdot \mathbf{B} &amp;= 4 \pi \rho_m \tag{2}\\ -\nabla \times \mathbf{E} &amp;= \frac{\partial \mathbf{B}}{\partial t} + 4 \pi \mathbf{J}_m \tag{3}\label{Eq:Faraday}\\ \nabla \times \mathbf{B} &amp;= \frac{\partial \mathbf{E}}{\partial t} + 4 \pi \mathbf{J}_e \tag{4}\label{Eq:Ampere} \end{align} $$ In particular, Faraday's law \eqref{Eq:Faraday} contains a minus sign that Ampere's law \eqref{Eq:Ampere} does not. This always struck me as odd because it's often said the fields are dual to each other (i.e. you can replace E with B and &quot;get the same result&quot;), but that requires a bit of mental recalibration to accommodate that minus. So I'm curious what the origin of that negative is and what it means. Are there any intuitive explanations for how to think about it? How to remove duplicate lines inside a text file? A huge (up to 2 GiB) text file of mine contains about 100 exact duplicates of every line in it (useless in my case, as the file is a CSV-like data table).  What I need is to remove all the repetitions while (preferably, but this can be sacrificed for a significant performance boost) maintaining the original sequence order. In the result each line is to be unique. If there were 100 equal lines (usually the duplicates are spread across the file and won't be neighbours) there is to be only one of the kind left.  I have written a program in Scala (consider it Java if you don't know about Scala) to implement this. But maybe there are faster C-written native tools able to do this faster?  UPDATE: the awk '!seen[$0]++' filename solution seemed working just fine for me as long as the files were near 2 GiB or smaller but now as I am to clean-up a 8 GiB file it doesn't work any more. It seems taking infinity on a Mac with 4 GiB RAM and a 64-bit Windows 7 PC with 4 GiB RAM and 6 GiB swap just runs out of memory. And I don't feel enthusiastic about trying it on Linux with 4 GiB RAM given this experience. What is wrong with this proof?    Is $\pi=4?$ I have a Google Apps for your domain account for my domain which we will call DomainA  I have since replaced that domain with DomainB which is set up in the Google Apps account as an alias.   Since I want to let DomainA expire at some point in the future, I would like to change my primary domain to DomainB in my Google Apps account with DomainA as an alias.   How can I do this? Apache SSL VirtualHosts on a single IP using UCC/SAN certificate I need to host several Apache virtual hosts with SSL from a single IP.  Now - I understand that because SSL wraps around the HTTP request, there's no way to know which host is being requested until a public key has been sent to the client first. This essentially breaks the possibility of SSL virtual hosts using a standard SSL certificate.  I have obtained a Unified Communications Certificate (UCC), otherwise known as a Subject Alternative Name (SAN) certificate. This allows me to serve the same certificate for multiple domains.  I would like this to be the certificate served by Apache for any SSL request - and then have Apache resolve the virtual host as usual, once the encryption has been established.  How should I configure Apache for this? I have tried to research how this can be done, but all I can find are quotes which say that it is possible, but no specifics:    wiki.apache.org/httpd/NameBasedSSLVHostsWithSNI     While Apache can renegotiate the SSL   connection later after seeing the   hostname in the request (and does),   that's too late to pick the right   server certificate to use to match the   request hostname during the initial   handshake, resulting in browser   warnings/errors about certificates   having the wrong hostname in them.   serverfault.com/questions/48334/apache-virtual-hosts-with-ssl     Incidentally, it is possible to have   multiple SSL-secured named virtual   hosts on a single IP address - I do it   on my website - but it produces all   sorts of warnings in the Apache logs,   and certificate warnings in the   browser. I certainly wouldn't   recommend it for a production site   that needs to look clean.   -David Jul 31 at 4:58   www.digicert.com/subject-alternative-name.htm     Virtual Host Multiple SSL sites on a single   IP address. Hosting multiple   SSL-enabled sites on a single server   typically requires a unique IP address   per site, but a certificate with   Subject Alternative Names can solve   this problem. Microsoft IIS 6 and   Apache are both able to Virtual Host   HTTPS sites using Unified   Communications SSL, also known as SAN   certificates.     Please help.