suppose $f(x,y)=\frac{|x|^a|y|^b}{|x|^c+|y|^d}$ and $a,b,c,d &gt;0$ how would you show that if $\frac{a}{c} + \frac{b}{d} &gt;1$ then $\lim\limits_{(x,y)\to(0,0)}\frac{|x|^a|y|^b}{|x|^c+|y|^d}$ exists and equals $0$?  I've been trying to use the squeeze theorem and set up an inequality but im really struggling.
Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas?
Link to the recommended job is broken in the mobile apps I'm using retextured snowballs and eggs for the different paintballs for two teams in a paintball map, so I can't replace the eggs with snowballs.  I have tried using /gamerule doMobSpawning false, but that still allows the chickens to spawn. I have also tried using /tp @e[type=Chicken] 0 0 0 on a repeating command block to quietly remove the chicken, but there's still a very noticeable frame where the chicken is visible. /effect @e[type=Chicken] minecraft:invisibility 10 1 false isn't fast enough, either.  I'm not using any plugins or mods.  tl;dr Without snowballs, how can I stop chickens from appearing for even a single frame. When I watch the news I've noticed that the Gallup polls for things like presidential elections have [I assume random] sample sizes of well over 1,000.  From what I remember from college statistics was that a sample size of 30 was a "significantly large" sample.  It was made to seem that a sample size over 30 is pointless due to diminishing returns. Why not perma-ban instead of suspend an account for 275 years? I know there are a lot of questions about very long suspension periods, but I came across a case that is quite perplexing. This user had many offenses to earn his long ban, despite being a very prolific and high-rep user with many valuable posts, but the suspension period assigned strains the definition of "suspension."     This account is temporarily suspended network-wide. The suspension   period ends on Mar 18 '(22)92 at 16:28.   I can see why you might want to suspend an account for several years, maybe even a decade, if you don't believe behavior will change. However, what is the point of "suspending" an account for 275 years? Is there a reason why you can't just make it a permanent ban instead of pretending it is a suspension?    On a tangential note, isn't the "temporary" in "temporary suspension" redundant?  Suspend: temporarily prevent from continuing or being in force or effect. I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.  It's a True/False question.  What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.  Thank you How can I find infinity focus on a kit lens with no markers? I have a kit lens which has no markers. What would be a recommended way to find infinity focus on this lens?  I plan to shoot objects at night -- not necessarily stars or the moon. So, adjusting the focus using the viewfinder may not be an option. Without solving the equation, show that all nine roots of $(𝑧 − 1)^{10} = 𝑧^{10}$ lie on the line $Re (𝑧) = \frac{1}{2}$ Problem: Without solving the equation, show that all nine roots of $(𝑧 − 1)^{10} = 𝑧^{10}$ lie on the line $Re(𝑧) = \frac{1}{2}$. I would like to say I tried something but I am actually clueless. I was thinking maybe I could use De Moivre's theorem but I am not sure how. Any hint would be appreciated. Are the WotC archives of d20 Character Optimization forum forever lost? I was looking for the Wizard Handbook but all the links were broken.  Somewhere on the handbooks of other classes (on other forums) I saw a      This was archived by WotC, here's the link   But the link leads to the welcome page of WotC (since it's broken).  Is there a way to get all that info back? Do you know where to get it from? The Wayback Machine did not archive the forums. How do you end with a single quote followed by a double quote? I have a paragraph as follows:  ``This is a quote. `Here is a quote within a quote.'''   But the formatting works out to be a double quote (") followed by a single quote (') instead of the other way around. How can I solve this problem? How can I run a program on startup, minimized? I just want Telegram to be run and I have added it to startup apps. The point is that I need it to be minimized. Any commands? Many of the Ringbearers (particularly those who carried it longer) used the word "precious" to describe it. Isildur described it as:     "But for my part I will risk no hurt to this thing: of all the works of Sauron the only fair. It is precious to me, though I buy it with great pain."   Bilbo himself used it at least once, when arguing with Gandalf about leaving it behind when he was leaving Bag End (this exchange and the lines were different in the movie, but the word is still there):     "Well, if you want my ring yourself, say so!" cried Bilbo. "But you won’t get it. I won’t give my precious away, I tell you."   And Gollum used it incredibly frequently, almost as if it was the ring's name. It was even his last words:     "Precious, precious, precious!" Gollum cried. "My Precious! O my Precious!"   So my question: Why "Precious"? It wasn't part of the inscription. It wasn't part of the poem that included the inscription. Was it somehow part of Sauron's crafting spell? Did Tolkien ever indicate why the Ringbearers would use that word in particular, other than that it sounded good? How do I prevent yum from updating a particular RPM? For certain reasons I have to keep a slightly older version of a particular RPM installed on my box (Fedora 11). The problem is that every time I do a package update yum (or one of its GUI equivalents) wants to update this RPM to the latest version, which is annoying - especially when I update it by mistake.  Is there a way to 'pin' this version so that yum knows to never try and update it? How do I use filenames that start with a dash "-" as command arguments? I am looking for a convenient way to set the background of tabular cells. I'd like to do two things:  (1) Set the background shaded in a color (2) Set the background hatched  I learned that the TikZ package provides means to do this but I am unable to come up with a solution that does not use matrices (such as here: ) but instead uses the tabular package.  Here's what I'd like to do:  \documentclass{article} \begin{document} \begin{tabular}{|l|l|l|} \hline   1 &amp; \setBGGray 2 &amp; 3 \\\hline   4 &amp; 5 &amp; 6 \\\hline   7 &amp; \setBGHatched 8 &amp; 9 \\\hline \end{tabular} \end{document}   The output should look like this:    Can someone please help me achieving this using tabular (or tabularx) and TikZ? MySQL convert date string to Unix timestamp Can the Universe be deterministic and still allow free will? I forgot the passcode for my iPad restrictions. How can I disable them? It's big and empty and I can't get up to the top, what should I do? I can't jump high enough, anti-gravity doesn't seem to do anything either.  My first tactic was to use the octopus summoning crown to create a pile of octopi that I could climb up, but when I teleported to the back of the room they followed me back, trapping me in the entrance. What is accountsd and why does it want to use the "login" keychain? Is there a way to do custom /say commands? I've looked it up, and apparently there is not a way to. But, there's an NPC in my map where I have it "talking" with a /say command. So, is there a way to make it say something like [(NPC Name)] (Text) without mods?