The question: $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(x+y)}{x+y}.$$ I tried this using $y=x$ path and $y=0$ path and both approach to the same value, $1$. My problem is to show that this value really approaches to one on any path  using the $\varepsilon$-$\delta$ definition.
I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :)
$\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$. I tried to assume that there is an isomorphism and to get a contradiction.. didn't succeed Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1? I was wondering why do some people use redundant axioms in definitions?  If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ab and 1. Then, you get it's commutative for free.   Why do we then assume it?   Also, is it even stronger. If we drop the condition that 1 is in R, then can we still deduce a+b=b+a? As clearly the argument just done falls apart if you haven't got 1 in R. Minimum safe spacing between lava and flamable materials? It seems that containing lava in solid glass, stone and iron blocks may not be sufficient to prevent wool from burning, even if the lava is completely encased.         After some delay, it seems that fire will jump onto the wool above the iron.   Are solid blocks more effective than airspace, or not? If so, Does it matter whether the corner block is filled in? What are the exact rules on distance that I need, and how can I prevent flammable blocks from being eliminated by nearby lava? Bonus: Does a flowing lava block behave differently than a lava source block when it comes to fire creation? How do I copy a file in Python?  I couldn't find anything under . Alternate moves, first one to roll a 6 wins: what's the probability of winning? If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton! I am looking for an adjective, similar to "holding a grudge" I am looking for an adjective that describes a person who is not willing to forgive or excuse someone's faults or wrong-doings from the past (other than "unforgiving"). For the ones who speak German, that word would be "nachtragend". I've been reading , and I would like to know why the Lasso provides variable selection and ridge regression doesn't.  Both methods minimize the residual sum of squares and have a constraint on the possible values of the parameters $\beta$. For the Lasso, the constraint is $||\beta||_1 \le t$, whereas for ridge it is $||\beta||_2 \le t$, for some $t$.  I've seen the diamond vs ellipse picture in the book and I have some intuition as for why the Lasso can hit the corners of the constrained region, which implies that one of the coefficients is set to zero. However, my intuition is rather weak, and I'm not convinced. It should be easy to see, but I don't know why this is true.  So I guess I'm looking for a mathematical justification, or an intuitive explanation of why the contours of the residual sum of squares are likely to hit the corners of the  $||\beta||_1$ constrained region (whereas this situation is unlikely if the constraint is $||\beta||_2$). How do I reset proxy in terminal to automatic if not connected via proxy I tried to reset proxy of the terminal by some commands but it doesn't happen and automatically switches back to this proxy 172.16.0.16 (which apparently was my college proxy).   I checked in my system settings.I don't understand why this is recurring. Please be comprehensive.Also I further would like to know how to bypass proxy server since I couldn't access any of the ubuntu repositories as they were blocked in my college's proxy settings as is Ubuntu's homepage.  Thanks for your time.    For sudo ls /etc/apt/apt.conf.d/        It displays a different set of options where proxy is not listed.I am on 12.10,if this should help any.I put a snap of the terminal after the above command has been entered. Can matter really fall through an event horizon? PHP: Static and non Static functions and Objects How to smooth camera animation to have fluid feel? When dealing with camera motion, its very sensitive to any movement discontinuities. Is there a way to easily achieve natural looking motion when dealing with complicated motion paths and timings?  Follow path animation feels too robotic unless I adjust all the keyframe easings carefully and this has to be re-done everytime the animation changes (the automatic bezier type just doesn't work). Azure DNS does not lookup SPF policies for certain domains How to prevent negative comments/downvoted being deleted by re-posting answers After Google Play Service update to version 13 I got an error Exclusive lighting Is (or will be) there any sort of light linking in Cycles, or a method to get a light to affect only certain objects that doesn't involve compositing two or more different render layers? Sort of what 3Dstudio does with the Include/Exclude methods. Tool to manage and/or make available a list of my publications on the web? How did the letter Z come to be associated with sleeping/snoring? What's the best way to protect your house from Endermen stealing your walls? Is there a convergent, alternating series that fails the AST? The alternating series test (AST) says, briefly, that if   $a_k&gt;0$ $a_k \geq a_{k+1}$, and $a_k \to 0$ as $k \to \infty$   then $\sum_k (-1)^k a_k$ converges.  This seems to be a one-way test (that is, if an alternating series fails the test, we don't know that it diverges).   document says so explicitly. It then gives an example of an alternating series which fails the AST. That doesn't prove the series is divergent (but it turns out to be divergent, anyway, by $n$th term test).  Is there a convergent, alternating series that fails the AST?  Of course, if the series fails condition 3, then it also fails the $n$th term test, and must diverge. And if it fails the first condition, then it's not strictly alternating, anyway.  So it must be a series that is not getting smaller (condition 2), but still converges.