Prove by using the formal definition of limit that:   $\displaystyle\lim_{x\to \infty  }\dfrac{1}{x^4 + x^2 + 5} = 0$  Can anyone please lend a hand here?
How would I go about proving this limit?  $\lim_{x\to\infty}\frac{1}{x^4+x^2+5}=0$  so far I have:  $|f(x) -L|&lt; ϵ$ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5} - 0| &lt; ϵ $ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5}| &lt; ϵ  ,\; x &gt; ∞$, assuming, $x &gt; 0$  taking the absolute value $\frac{1}{x^4+x^2+5} &lt; ϵ$  ${x^4+x^2+5}$ > $\frac{1}{ϵ}$  i am not sure whether this is correct, but what would be the next step?
Which types of particles are affected by the wave-particle duality? If we take the double slit experiment as a way of demonstrating the wave-particle duality, which types of particles would show an interference pattern?  For example, I know that electrons show such a pattern. But do protons, too? What about atoms? Where is the boundary between "wavey particles" and "classical particles"? It's easy to set inline CSS values with javascript. If I want to change the width and I have html like this:  &lt;div style="width: 10px"&gt;&lt;/div&gt;   All I need to do is:  document.getElementById('id').style.width = value;   It will change the inline stylesheet values. Normally this isn't a problem, because the inline style overrides the stylesheet. Example:  &lt;style&gt;    #tId {       width: 50%;    } &lt;/style&gt;  &lt;div id="tId"&gt;&lt;/div&gt;   Using this Javascript:  document.getElementById('tId').style.width = "30%";   I get the following:  &lt;style&gt;    #tId {       width: 50%;    } &lt;/style&gt;  &lt;div id="tId" style="width: 30%";&gt;&lt;/div&gt;   This is a problem, because not only do I not want to change inline values, If I look for the width before I set it, when I have:  &lt;div id="tId"&gt;&lt;/div&gt;   The value returned is Null, so if I have Javascript that needs to know the width of something to do some logic (I increase the width by 1%, not to a specific value), getting back Null when I expect the string "50%" doesn't really work.  So my question: I have values in a CSS style that are not located inline, how can I get these values? How can I modify the style instead of the inline values, given an id? How do I evaluate several n-th derivatives of a function at one point? Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint, storm drains clogged with moss, dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint; storm drains clogged with moss; dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights. Walls covered with cracks and peeling paint. Storm drains clogged with moss. Dumpsters overflowing with bags, cans, and rotten trash.  Or maybe the three of them are orthographically correct in their respective contexts? EDIT (based on dockeryZ's answer): I think second example is right, regardless of the context. Explanation: . Eclipse was working as good as anything on 14.04. I did a clean install of 16.04 and installed Eclipse. But it runs a Java program only once after which it just hangs during subsequent attempts before I remove and reinstall it. I have tried many things, since in the beginning I needed CDT as well. Right now, I just want to remove everything related to eclipse and do a fresh install (remove all the configs and dependencies). Please note that I used Eclipse installer to install Eclipse and the eclipse icon image is also not appearing (the default ubuntu ? is appearing). I am trying to solve a 2 part question, the first part is to expand $\sqrt{1-x}$ up to and including the term $x^2$ which I did.  This gives me $1-(0.5)x-(0.125)x^2 + ...$  However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $\sqrt 7 = 10837/4096$. How do i do this part?  EDIT (After Hint Given By SimpliFire):  $$ \sqrt{1-\frac1{64}}=\frac{\sqrt{63}}8=\frac38\sqrt7 $$ $$ =\frac83\sqrt{1-\frac1{64}}\;=\sqrt7 $$ $$ =\frac83\left(1-\frac12\left(\frac1{64}\right)-\frac18\left(\frac1{64}\right)^2\right)\; $$ $$ =\frac83\left(1-\frac1{128}-\frac1{32768}\right)\; $$ $$ =\frac83\left(\frac{32768}{32768}-\frac{256}{32768}-\frac1{32768}\right)\; $$ $$ =\frac83\left(\frac{32511}{32768}\right)\; $$ $$ =\frac{260088}{98304} $$ $$ \frac{10837}{4096}\;\approx\sqrt7 $$  I'm not sure if I can present it like this, if it is okay i will post it as my answer :X keyboard hard remap keys? I accidentally set an object as camera, how do I undo this? Blank screen after installing nvidia restricted driver Already awarded bounty refunded on question migration? In March 2011 I offered a 500 bounty on one of my StackOverflow questions () and awarded it to Filip de Vos.   This question was later migrated to Database Administrators and has the following messages on it.     migrated to dba.stackexchange.com by Will♦ Dec 28 '11 at 12:25      This   question belongs on our site for database professionals who wish to   improve their database skills and learn from others in the community.   .     deleted Dec 31 '11 at 1:01   I appear to have been   on whereas Filip .  Is this a bug? Definitely doesn't seem fair or correct to me. The question still shows up in both our bounty tabs despite the migration. I would have hoped that the previously awarded bounty reputation points could still be honoured in this situation. Order of a element on a Symmetric Group I have a Samsung 960 Pro SSD with Ubuntu 18.04  is it safe to put the swap file on there or would it hurt it in the long run,  I heard some people saying that new SSDs don't suffer from that issue anymore, is this true? For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous. I was just trying to solve an exercise in Munkres' Analysis on Manifolds to get better at continuous functions and topology in $\mathbb{R}^{n}$, and I got stuck. Here it is:      Let $X = A \cup B$, $A$ and $B$ are subspaces of $X$. Let $f:X \rightarrow Y.$ Suppose that the restricted functions $f_{|A}:A \rightarrow Y$ and $f_{|B}:B \rightarrow Y$ are continuous. Show that if both $A$ and $B$ are closed in $X$, then $f$ is continuous.    Our definition of continuity is:  $f$ is continuous at $x \in X$ if for each open set $V$ of $Y$ containing $f(x)$, there is an open set $U$ of $X$ containing $x$ such that $f(U) \subset V$. $f$ is continuous if it is continuous at each point of its domain.  What I've tried so far:  Suppose $A$ and $B$ are closed in $X$. Suppose $A \cap B = \emptyset$. Then $X -A$ is open in $X$. But $X - A = B$ which is closed in $X$, which is a contradiction. Hence, $A \cap B \ne  \emptyset$. I'm not sure if this will be useful in the problem or not. If $x \in X$, then it's either in $A$ or in $B$. Suppose without loss of generality that it's in $A$. Then for all open sets $V$ of $Y$ s.t. $f(x) \in V$, $\exists$ an open set $U_{1}$ of $A$ s.t. $U_{1} \subset A, a \in U_{1}$ and $f(U_{1}) \subset V$. This is almost what we want, except that we want to find an open set $U$ of $X$ s.t. $U \subset A, a \in U$ and $f(U) \subset V$. This is because $U_{1}$ being open in $A$ does not necessarily mean that it's open in $X$ because $A$ is closed in $X$. (i know how to expand this argument out if needed) My idea now is to try to find a subset $U$ of $U_{1}$ such that $U$ is open in $X$. We have two cases:  (1) $U_{1}$ does not intersect the boundary of $A$. (2) $U_{1}$ intersects the boundary of $A$. Now I'm not sure what to do. I'm wondering if it would help to investigate the relationship between Bd(A) and Bd(B).. In any case (no pun intended), any help/advice would be greatly appreciated. :)  Sincerely,  Vien ASP.NET Website's BIN directory and references PHP not working after upgraded to Ubuntu 17.10 After I upgraded to Ubuntu 17.10, PHP totally stopped working. Everything works fine in Ubuntu 17.04. The code for info.php as follows does not work:   &lt;?php  phpinfo(); ?&gt;   It is not showing PHP information, it only shows the source code like above. "The resource cannot be found." error when there is a "dot" at the end of the url Toggle wireframe mode with hotkey How do I toggle in and out of wireframe mode with my keyboard? Is there a hotkey for this?  I'm a beginner (obviously). I thought I could find this in the "view" menu, but it either doesn't exist or I have horrendous eyesight. Thanks! Can I use my iPhone as a "remote" for my iPad? Can I travel in UK before switching to a study visa? As I read in general visitor visa Terms &amp; Condition, I would only be granted to stay up to 6 months in UK. However I am planning to apply for General Student Visa to further my studies. Would it be possible for me to enter UK as visitor for first 6 months/less before I apply for student visa in the same year if I want to travel and study on the same year? Do you need to disavow on [HTTPS SSL] after performing a disavow on [HTTP] I have a client which disavowed a few thousand dirty links from his site profile back in February 2015. Recently the site was switch from HTTP to HTTPS (SSL enforced).   Google recommends that you add all variations to the Google Console (Webmaster Tools).  The site has a total of 4 variations, eg:    http://www.example.com (Previously Preferred) http://example.com https://www.example.com (Now Preferred) https://example.com   Question:  Since technically this is a new property in Google Webmaster Tools, do you need to submit the same disavow file added previously back in 2015?