I am trying to solve the following limit:  $\lim_{(x,y) \to (0,0)} \frac{x^4y^4}{(x^2 + y^4)^3}$  (This is a more challenging problem from Folland Calculus, it seems).  I am pretty sure this limit does not exist (however, this is just a guess, and I am not 100% sure.)  I am trying to approach via the paths y = mx and x = my but... they don't seem to work. I have tried simpler cases such as x = 0 and y = 0.   Any help appreciate.
How to check if this limit exists: $$\lim_{(x,y)\rightarrow (0,0)}{ \frac{x^4y^4}{(x^2 + y^4)^3}}$$  Can I convert it to polar form ? $$\frac{r^8 (\cos^4 (\theta)\sin^4(\theta)}{r^6 (\cos^6 (\theta) + r^6 \sin^{12} (\theta))} = \frac{r^2}{\cos^6(\theta) + 0} = 0$$
80/90s UK children's drama series about an "alien" boy with radio weapon I've been trying to track down a live-action children's TV series (or short story perhaps) from the 1980s or 1990s, probably on CBBC or CITV.  A few people I've spoken to IRL have suggested The Boy from Space or Sky, but I'm pretty sure it's neither as a lot of the details don't seem to match.  The story was set on Earth (possibly England or Wales) and was about an "alien" boy who was being pursued by a group of people, though I don't remember who they were.  Somehow he encountered siblings, a girl and a boy, and made friends with them. I think they lived on a farm with their parents, as I remember quite a large barn or other building, and they hid(?) the boy in a car which was inside or next to this building. He also slept in this car with a dog (which might have been a German shepherd).  The "alien" boy had a portable "radio" (all silver, very 1980s) that repelled his pursuers by emitting a high-pitched sound when he turned a dial on it, and he used it when out and about with the other children whenever they encountered a pursuer. I seem to remember lots of hills and valleys, lots of green.  In the end it turned out he wasn’t really an alien, but a human from the far future. He was pretending to be an alien as he was from 20000 years into the future and humans' appearance had changed by then, but he had the ability to make himself look like current-day humans. When he was in his car/bed he reverted to his normal appearance, so he would hide under the blankets until he was able to assume current-day human appearance.  In the finale, it was revealed that he was never in any danger – the pursuers were testing him as part of a kind of coming-of-age thing. I remember a scene on a spaceship or something where his elders/parents where shown telling him that he was safe all along and that he "passed".  I also seem to remember them referring to "ancestors" and making sure they didn't find out, so the either the girl or boy could well have been a direct ancestor. Can a child enter the UK on an accompanied visa and then leave the country alone? My daughter who is just under 18 has a child accompanied visa to the UK. I am traveling with her to the UK for a few days. We return from UK on the same day except that she returns to India and I am heading to San Francisco an hour later on the same day.   Is that okay or will we have an issue at immigration into the United Kingdom? What are some scenarios where FTL information transfer would violate causality? I've always heard people saying, "Faster than light information transfer can't happen because it would violate causality!  Effects can precede their causes!"  However, I'm trying to think of a situation where this would happen.  I'm sure it has something to do with relativity or something like that.  I sort of understand that people going faster perceive time slower.  Can someone help illuminate this for me by showing a scenario where causality is clearly violated due to FTL information transfer? The infinite product of the sines of all positive integers is zero I tried to prove that $\prod\limits_{k = 1}^\infty  {\sin k = 0}$  Define $$P(m)=\prod\limits_{k = 1}^m  {\left|\sin k\right|}$$  then I take the logarithm of both sides $$\log P(m)=\log \prod\limits_{k = 1}^m  {\left|\sin k\right|}$$  which can be written as $$a_m=\log P(m)=\sum\limits_{k = 1}^m  {\log\,\left|\sin k\right|}$$ for any $k\in\mathbb{N}$ we have $0&lt;\left|\sin k\right|&lt;1$, so all terms of the sum above is strictly negative.  I say that the sequence diverges: $\left\{a_m\right\}_{m\in\mathbb{N}}\to\,-\infty$  This is a delicate point on which I have some doubts.  I know that  $$\int\limits_0^\pi  {\log \left|\sin x\right|{\text{ d}}x}  =  - \pi \log 2$$ and that the function $\log \left|\sin x\right|$ is periodic with period $\pi$,  so the integral from 1 to $\infty$ diverges to $- \infty$. Then the series diverges as well for the integral test for convergence.  As $\log P(m)\to\, -\infty$ we can say that $P(m)\to\,0$ as $m\to\,\infty$  Therefore $$\mathop {\lim }\limits_{m \to \infty } \prod\limits_{k = 1}^m {\left| {\sin k} \right| = \mathop {\lim }\limits_{m \to \infty } \prod\limits_{k = 1}^m {\sin k = \prod\limits_{k = 1}^\infty  {\sin k = } 0} }.\quad \blacksquare$$ The motivation for this question is in my answer to this post :       (1) If I take $A$, $B$ and $C$ three groups such that $A\times B$ is isomorphic to $A\times C$ then do we have $B$ is isomorphic to $C$ ?   In the post a counter example is given when $A$ is an infinitely generated group. However I cannot get a counter-example when $A$, $B$ and $C$ are assumed to be finite groups. Note that there is no reason a priori if $\phi$ is an isomorphism from $A\times B$ to $A\times C$ that $\phi(A\times\{1_B\})=A\times\{1_C\}$.   In the case of finite groups it is true because of the Krull-Schmidt theorem : .  A slightly more general question is the following :     (2) If $B$ and $C$ are two finitely generated groups, can I always find a finitely generated group $A$ such that $A\times B$ is isomorphic to $A\times C$?   If $A$ is not assumed to verify any property but to be a group we can take :  $$A=B^{\mathbb{N}}\times C^{\mathbb{N}} $$  Furthermore replacing :$ \prod_{\mathbb{N}}$ by $\bigoplus_{\mathbb{N}}$ (or more likely the restricted product for groups but I am not sure of a well known notation, it is the subgroup of the product with all but a finite number components being trivial) will give you the same result for countably infinite groups.   $\underline{\text{Edit}}$ : To be clear about what I am asking. (1) is kown to be true for finite groups whereas (2) is known to be true when we are considering groups without restriction (even just for countably infinite groups). Let us pretend we are working on the smaller class of groups : the finitely generated groups. Is there any counter example to $(1)$? Is there any counter example to $(2)$?   $\underline{\text{Edit 2}}$: Here is a partial answer, from the classification of finitely generated abelian groups $(1)$ is always true if all groups are assumed to be abelian finitely generated groups. Now assume that $A\times B=G=A\times C$. Then you have (derived groups) :  $$D(G)=D(A)\times D(B)=D(A)\times D(C) $$  So the abelianizations verify :  $$A^{ab}\times B^{ab}=A^{ab}\times C^{ab}$$  Because we are now in the realm of finitely generated abelian groups we get :  $$B^{ab}=C^{ab} $$  So $(2)$ cannot be true in general for instance you can take $B:=\mathbb{Z}$ and $C=\mathbb{F}_5$, those two groups have different abelianizations.   The new question can be found here : How to run scripts without typing the full path? Correctly indexing multiple domains with same content in Google and others How to pronounce $\mathcal{E}$? How can I bind arguments to a Python method to store a nullary functor for later invocation?  Similar to C++'s boost::bind.  For example:  def add(x, y):     return x + y  add_5 = magic_function(add, 5) assert add_5(3) == 8 I want to fill a part of a picture with more or less evenly and randomly distributed circles which should not overlap. It would be okay if circles are only partly inside the shape.  The density should be like what you see on this page  with values 80-100. (But I do not need so many circles, 30 should be enough, I could repeat the pattern).   I cannot use a grid and "wobble" the circles a bit: The circles are too large and it simply does not look randomly.   Currently I am thinking about a random attempt: Get some random point, measure its distance to existing circles, either paint or throw the point away. Try 100 times and hope that you get the right number of circles (and that it does not take too long). But perhaps someone has a better idea.   The example draws the circles simply manually:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{backgrounds} \begin{document}   \begin{tikzpicture}[framed,gridded,radius=0.5cm]  \draw[red](0,0) rectangle (5,5);   \draw (1,1) circle;   \draw (2.3,1.1) circle ;   \draw (4.5,0.8) circle ;   \draw (5.1,1.8) circle ;   \draw (0.4,3.3) circle ;   \draw (2.1,2.8) circle ;   \draw (3.8,3.5) circle ;   \draw (4.8,4.2) circle ;   \draw (0.8,4.9) circle ;   \draw (2.1,4.1) circle ;   \draw (3.8,2.0) circle ;   \draw (3.5,0.6) circle ;   \draw (3.0,5.0) circle ;      \draw (4.1,5.1) circle ;             \draw (0.9,2.1) circle ;  \end{tikzpicture}  \end{document} How to use triangle inequality to establish Reverse triangle inequality I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$ .   I have tried to represent  $||a|-|b||$ as  $||a|+(-|b|)|$ , and then get $||a|+(-|b|)| \leq |a|+|-|b||$ , but that isn't leading me anywhere given $|a-b| \leq |a|+|b|$. Year-Round Hats for Fun and Profit "Software Updater" & "Software & Updates" not working I havent used my Ubuntu for quite a long time.  Today, when I booted Ubuntu (14.04 LTS), the software updater and Software &amp; Updates (from Settings) are not working.  The software updater's window will show up for around 1 sec, but it will close automatically. Software &amp; Updates is never shown, no matter how many times I click the icon. Note that neither of them show any error message.  Those 2 features were working perfectly the last time I used Ubuntu. I want to do some system upgrade and update.  For now, I use sudo apt-get update &amp;&amp; sudo apt-get upgrade. This command runs very fast, meaning (I think) my Ubuntu is not updated properly. The update from software updater always has a big size hence the download should be slower.  What should I do?   Can I just uninstall and reinstall both of them? If I can, how to do it?  Just to clarify, I don't want to upgrade my Ubuntu version (I want the LTS one), I just want to upgrade the software/packages I used.  Thanks for your time. Do I need to play previous Witchers before Witcher3 I'm thinking about buying The Witcher 3 and I know it's heavily based on a story/narrative.   For the story/narrative to make sense, do I need to play the previous 2 Witcher games - or are the stories for each title independent of one another? I am solving the TOEIC workbook, by the way, I faced conflict that doesn't match meaning I knew.  The problem is     The newest branch of the library is located in/on Pine Street.   I think either 'in' or 'on' have the same meaning. But the commentary said that 'be located in' is mainly writing in front of building name and city name.   I don't know what is correct? Calculating CR of monsters that do not deal calculable damage PHPUnit skips all tests Xcode free space requirement I am trying to update Xcode to 8.0. My OS is 10.11.6 El Capitan. However I am getting:     We could not complete your update   There is not enough disk space available to install the product.   I have got 14.22 GB free space. How much do I need?  UPDATE: I do not know the exact figure, though I did manage to update with 17.21 GB free space later on. Note: This is a canonical post created so that we can close the many similar questions as duplicates as discussed in .   We have created a logo for a client and they love it. We've already started preparing the materials that will feature the new logo.   However, after we had created the logo, we found a logo for another company that looks eerily similar. We did not intentionally create the new logo to look like the one of the other company, in fact we didn't even know it existed beforehand, but it does regardless.  What should we do? Is this plagiarism? Can our client use the logo we have created? Should they? Or do we need to come up with another logo that's more different? For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$ For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one element and the rest consist of two elements.  Solution  True. To show that $\equiv$ is reflexive we need to show that $\forall x\in\mathbb{R} :x\equiv x$. Let $x\in\mathbb{R} , x\equiv x \mbox{ if } x^{2}=x^{2}$, which is obvious.  $[x] ={y\in\mathbb{R} |x\equiv y} =[0]={y\in\mathbb{R}|0\equiv y}={0}$. Hence $y^{2} =0^{2} =0$ which implies that $y=0$.  Can anyone please give me correct answer to this question.