Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$ using the  definition of continuity.  My attempt:  Need show that $\forall {\epsilon &gt; 0},\exists {\delta &gt; 0}$ s.t $\forall {x \in R },\ |x - (-1)| &lt; \delta \implies |f(x) - f(-1)| &lt; \epsilon.\\ |f(x) - f(-1)| = |\frac{x}{x^2 + 1} - \frac{-1}{2}| = |\frac{2x - (-1)(x^2 + 1)}{2(x^2 + 1)}| = |\frac{2x + x^2 + 1)}{2(x^2 + 1)}| = |\frac{(x+1)^2}{2(x^2 + 1)}|\\ |\frac{(x+1)^2}{2(x^2 + 1)}| &lt; |\frac{(x+1)^2}{(x^2 + 1)}| &lt;  |(x+1)^2| \text{ since } (x^2 + 1) \text{ is at least 1}\\ |(x+1)^2| = |x + 1|^2 &lt; \epsilon \\  |x + 1| &lt; \sqrt{\epsilon}$  Therefore set $\delta = \sqrt{\epsilon} \text{ and } \forall {\epsilon &gt; 0}, \ \forall {x \in R }, |x - (-1)| &lt; \sqrt{\epsilon} \implies |f(x) - f(-1)| &lt; \epsilon$  is this a valid way of proving this? Or is there a better way?
Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the  definition of continuity.   I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".  Definition of continuity; for $f:A \to \mathbb{R}$     $$\forall \varepsilon &gt;0 \ \exists\delta&gt;0\ \forall x \forall x_0 \in A:|x-x_0|&lt;\delta \Rightarrow |f(x)-f(x_0)|&lt;\varepsilon$$   So from what I understand, the procedure for epsilon-delta proofs are as follows: 1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|&lt;\varepsilon$ 2. Write $x$ in terms of $\varepsilon$ 3. Then plug $x$ into $|x-x_0|&lt;\delta$ 4. Find $\delta$ in terms of $\varepsilon$. Done    At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|&lt;\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?  Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing...
How to handle a deliberate duplicate question ssh master - execute su remotely I have a server S that I need to connect to.  I'm connecting to S as user asdf with control master and after that I perform some stuff:  ssh -N -f -M -o ControlPath=$SSHSOCKET ${USER}@${IP} -p ${PORT} -i id_rsa ssh -to ControlPath=$SSHSOCKET ${USER}@${IP} -p ${PORT} "su -c whoami &amp;&amp; hostname &amp;&amp; exit"   It works as expected, but second line asks me  for password.  I am trying such fix:  echo 'qwerty' | ssh -t -to ControlPath=$SSHSOCKET ${K3_USER}@${IP} -p ${PORT} "su -c whoami &amp;&amp; hostname &amp;&amp; exit"   but it gets even worse:  muxserver_accept_control: tcgetattr: Inappropriate ioctl for device tcgetattr: Inappropriate ioctl for device Password:   For me, it's mandatory to ssh as asdf first. I also can't make any other ssh control master connections. I also can't modify any config on the target machine.  I just need to switch to root via ssh somehow. Any ideas ? How to restore the down arrow icon on the Downloads folder in the home directory? I removed the original Downloads folder in the home directory and created a new one with the same name, but the down arrow icon on the new Download folder now disappeared. Could someone please tell me how to restore that icon?   In addition, from the image above we see "Downloads" disappears also in "Places" category in the left panel. I think it should be there originally. If after restoring the down arrow icon, the "Downloads" is still not in "Places" category, I would also like to ask you to help me put "Downloads" back to "Places". Thank you.  PS: I am not using any theme. Orbiting a specific point - Orbiting Camera How would I create a camera that with the mouse, it allows you to orbit around a specifc position, with the camera always facing that point? I would also like it to be able to always be at a constant distance from the point, and be able to orbit around it in every direction.   Any ideas? Suppose $G$ is a group and $K\lhd H\lhd G$ are normal subgroups of $G$. Is $K$ a normal subgroup of $G$, i.e. $K\lhd G$? If not, what extra conditions on $G$ or $H$ make this possible?  Applying the definitions, we know $\{ghg^{-1}\mid h\in H\}=H$ and $\{hkh^{-1}\mid k\in K\}=K$, and want $\{gkg^{-1}\mid k\in K\}=K$. Clearly, the best avenue for a counterexample is if $gkg^{-1}\not\in K$ for some $k\in K$ and $g\in G-H$.  If no such element exists, $\{gkg^{-1}\mid k\in K\}\subseteq K$ implies $\{gkg^{-1}\mid k\in K\}=K$ because if $k'\in K$, $gk'g^{-1}=k\in K\Rightarrow k'=g^{-1}kg\in\{gkg^{-1}\mid k\in K\}$. "+-=,;..." math characters in DejaVu Sans (with mathspec) the following MWE  \documentclass{minimal} \usepackage{mathspec} \setmainfont{DejaVu Sans} \setmathsfont(Digits,Latin,Greek){DejaVu Sans} \setmathrm{DejaVu Sans} \begin{document} \(\mathrm{e}^{\mathrm{i}\pi}=-1\) = -1 \end{document}   typesets with xelatex into    How can I turn the = and - signs in math mode to DejaVu Sans ? Pdf of the square of a standard normal random variable Import error when opening empty notebook MySQL Database User: Which Privileges are needed? The short installation instruction for WordPress () state that:     Create a database for WordPress on your web server, as well as a MySQL user who has all privileges for accessing and modifying it.   While setting up a new blog professionally I was wondering how that maps to what the MySQL database user privileges/permissions configuration offers me:   Data: SELECT, INSERT, UPDATE, DELETE Definition: CREATE, ALTER, DROP Extra: INDEX More:  LOCK TABLES REFERENCES CREATE TEMPORARY TABLES CREATE VIEW SHOW VIEW CREATE ROUTINE EXECUTE ALTER ROUTINE    I'm pretty sure for the first three groups, I named them Data, Definition and Extra here. But what about the others below the More entry? Normally I would say, those are not needed, but I would like to get a second opinion. What are the difference between the following sentence?    I ate apples. I have eaten apples.    When should we use simple past tense? When should we use perfect past tense? Prepositional Phrase - confused - at or in Is Feyerabend confusing discovery and justification when he criticizes the scientific method? I am reading Feyerabend's "Against Method", where he uses Copernicus's (and Galileo's confirmation) discovery of the fact that the Earth orbits around the Sun and other examples to show that irrational approaches can lead to legitimate scientific progress. He then claims that no unified epistemic principle or principles can be used to define science, and that science is best characterized by anarchy.   Is he confusing the way scientific truth is discovered with the way scientific truth is epistemically justified?   Even if it is true that Copernicus, Galileo and others used irrational methods to arrive at valid scientific discoveries, why does that affect science's special epistemic status compared to other fields of inquiry? Gnome-shell seems to like to leave one monitor as "fixed" when you switch workspaces.  Additionally, there is no preview in the Activities view of my secondary monitor.  Also, on the second monitor, where there should be either empty desktop space, or a copy of the panel, I just have a white bar.  How can I get gnome-shell to treat my secondary monitor like the primary one? London Heathrow airport - do I need a transit visa? I am traveling from the US to India via London Heathrow on November 6. I will be at LHR on 7th at 6:20 am and departing at 9:25 pm. I'm an Indian national and have a valid US visa.  Do I need a transit visa to stay airside?  I have been scared that British Airways have stopped a few people from boarding flights (no details just rumors).  Do I need a transit visa to stay landside? I want to do some sightseeing. What are the chances of getting past officials using transit without visa concession?  Real stories will help a lot to calm the butterflies in my stomach. When I click the bounty info icon on the iOS app. The added bounty description and reason is not shown.  For example: Do all accepted students in top-tier engineering programs get invited to interview before getting admitted? removeClass and fade animation MySQL GUI Tools Question about the order of the product of two elements in an abelian group My reference for learning abstract algebra says that in an abelian group $G$ the product of two elements (say $a$ with order $n$ and $b$ with order $m$) has order $mn$ if $gcd(n,m) = 1$.   What I don't understand is where the $gcd(m,n)=1$ comes into use in proving this fact? For example, $ a^{mn} = (a^n)^m = (e)^m = e$ (identity element). A similar argument for $b^{mn}$ together with the fact that $G$ is abelian is enough to show that $(ab)^{mn} = e$. From here, how do I conclude for an integer $r &lt; mn$ that $(ab)^r \neq e$  Also, are there any other conclusion that can be drawn if $gcd(n,m) \neq 1$ ?  Thanks for any feedback in advance ! 3rd iterate of a continuous function equals identity function