I've tried plotting the graph of the function in Desmos and WolframAlpha, and true to the limit, the function tends to 1.  I would assume by the product rule of limits, that  $\lim_{x \to 0} x = 0$, and hence the whole lot should be zero, too, but apparently not, as the behaviour of the floor function somehow makes the function tend to 1.   Or does it behave like $\lim_{x \to 0} \frac{x}{x} = 1 \mid x \in \mathbb{R}\backslash \{0\}$—which is much more straightforward to deduce? If so, why? I'm a little confused.
Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$  I'm trying to solve it by using the squeeze theorem but I'm stuck.  I'm looking for a function $g(x)$ such that $g(x) \leq x\lfloor\frac{1}{x}\rfloor \leq x(\frac{1}{x}) = 1$  Can anybody give me a hint?
I need to prove that the language $\mathscr L=\{0^n \big|\text{n is the square of a natural number}\}$ is non regular using the pumping lemma   My try:  $\mathscr L=\{\overbrace{\epsilon}^{0^2},\overbrace{0}^{1^2},\overbrace{0000}^{2^2},\overbrace{000000000}^{3^2},\overbrace{0000000000000000}^{4^2},\dots\}$  Suppose that $\mathscr L$ is regular so $\exists$ a word '$x$' with length of at least $n$   $|x|\geq n $ such that  $(1)\,\,\,|uv|\leq n$  $(2)\,\,\,|v|\geq 1$  $(3)\,\,\,uv^iw \in \mathscr L$  Now, let as choose the word $\color{blue}{x=0^{n^2}}$  $|x|\geq n$ so we can use $(1)-(3)$  $x=\underbrace{000000\dots}_{n^2\text{ times}}$  $x=uv0^kw=\underbrace{uv0^k}_{\text{n zeros   }}\overbrace{w}^{n^2-n\text{ zeros }}$  Let as choose $\color{blue}{i=??}$      I'm difficult choosing the word '$\color{blue}x$' and choosing  $\color{blue}i$ to come to contradiction with the lemma Maximizing expected value of coin reveal game I was asked this question today in an interview and am still not sure of the answer. The question is as follows.   Part one:  Say I have flipped 100 coins already and they are hidden to you. Now I reveal these coins one by one and you are to guess if it is heads or tails before I reveal the coin. If you get it correct, you get $\$1$. If you get it incorrect, you get $\$0$. I will allow you to ask me one yes or no question about the sequence for free. What will it be to maximize your profit?  My approach for this part of the problem was to ask if there were more heads than tails. If they say yes, I will just guess all heads otherwise I just guess all tails. I know the expected value for this should be greater than 50 but is it possible to calculate the exact value for this? If so, how would you do it?   Part two: Same scenario as before but now I will charge for a question. I will allow you to ask me any amount of yes or no questions as I go through this process for $\$1$. What is your strategy to maximize your profit?  I was not sure about the answer to this part of the question. Would the best option be to guess randomly? I think the expected value of this should be 50. I am not sure about the expected value of part one but if it is greater than 51, I think I could also use that approach. Anyone have a good idea for this part? Excel to CSV with UTF8 encoding I'm a student who uses the laptop all the time and I have a problem with Wifi on the laptop. I don't really know what it is. I can describe the problem tho. For example, i closed my laptop last night, so it goes in suspend mode. I turn it on this morning and my wifi doesn't want to connect until I restart the laptop. It says that it's connecting and in Wifi settings the circle animation is spinning, turning Wifi off and on doesn't do anything. Here's what happens when I check to see my network info: sudo lshw -C network   *-network                         description: Wireless interface        product: Wireless 3160        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:02:00.0        logical name: wlp2s0        version: 83        serial: 00:1e:64:e6:1c:93        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless        configuration: broadcast=yes driver=iwlwifi driverversion=5.4.0-52-generic firmware=17.3216344376.0 ip=192.168.0.14 latency=0 link=yes multicast=yes wireless=IEEE 802.11        resources: irq:129 memory:df100000-df101fff   *-network        description: Ethernet interface        product: RTL810xE PCI Express Fast Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: enp3s0        version: 07        serial: f8:ca:b8:52:72:49        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 firmware=rtl8106e-1_0.0.1 06/29/12 latency=0 link=no multicast=yes port=MII        resources: irq:17 ioport:d000(size=256) memory:df000000-df000fff memory:d0000000-d0003fff  Also I have to mention that my Linux is doublebooted alongside Windows 10, but I don't think that makes any difference. Help would be much appreciated, I wouldn't want to miss my classes because of this hahah Thanks! Making an Emission shader emit a different colour of light than the colour assigned to the object? Update:  Thanks for all the feedback and bug reports! I am locking this thread so people don't keep answering it when we no longer monitor it.  If you see a bug or want to raise a feature request, please do so in the usual manner, as a new question.    Some of you may have noticed that  the  and  pages that we have been talking about for a while now on .  Most? Well, yes, the new design is dependent on a SE network-wide update (a new base css framework) that the design team have been working on. This is an ongoing effort and we will be launching the new pages as sites are moved to the new design.  You've all been hugely helpful already - much of what we love most in here came from  and  you gave us on Meta.     But it's possible that it's not quite perfect yet. If you find any bugs and have any feedback to give, please post these as answers to this post.  Just post one issue per answer - that would help us focus on specific items one at a time. How do I force TeX to completely use the 1st column? How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Is it possible to prevent DHCP service on SBS 2003 from shutting down due to rogue DHCP server? I've just started learning Lisp and I can't figure out how to compile and link lisp code to an executable.  I'm using clisp and clisp -c produces two files:   .fas .lib   What do I do next to get an executable? AppleScript to read all notifications in the Notification Center OSX 10.10 Yosemite It seems like the following AppleScript is no longer working in the latest version of the OS X 10.10 (Yosemite): on run     tell application &quot;System Events&quot;         tell process &quot;Notification Center&quot;             set theseWindows to every window             display notification (count of theseWindows)         end tell     end tell end run  This were supposed to get all notifications listed in the Notification Center. However, the count attribute always returns 0. Does anyone know how to retrieve all notifications in the Notification Center different than the script above? cron job schedule At what time will the upcoming Ubuntu release be made available? Every six months a new Ubuntu version will be released. Does anyone know at which time this will be? $\limsup$ and cluster points Let $(x_n)$ be a sequence in $\mathbb R$. We call $y$ a cluster point of $(x_n)$ iff for every neighborhood $N$ of $y$ there are infinitely many $n$ such that $x_n \in N$. Let $C$ denote the set of cluster points of $(x_n)$.   By definition, $\limsup_{n \to \infty} x_n = \inf_n \sup_{k \geq n} x_k$.  Can you show me how to prove $\limsup_n x_n = \sup C$? Thanks. How can I check if a package is installed (no superuser privileges)? At our university we can get almost any ubuntu package installed we want, but we are not superusers ourselves (we need to request packages being installed).  With some libraries it is not always easy to know whether the package is already installed or not. Is there a simple way/command to check this? If the plural of ‘man’ is ‘men,’ shouldn’t the plural of ‘German’ be ‘Germen’? What makes these two words so different that 'man' is changed to 'men', but 'German' is changed to 'Germans'? Recursively delete empty directories in Windows How to validate an email address in JavaScript Analyze a circuit with two diodes and two voltage rails I want to find \$I\$, \$I_{D2}\$ and \$V\$ in this circuit. The diodes are made of Silicium. I'm a beginner and normally I would assume the Diode bias and then apply KVL to see if it is true. However, I've only encountered circuits where the wire forms a loop.    How am I supposed to apply KVL in this case? If voltage is the difference in potential between two points, how can I find the voltage of V if one point is outside of the circuit, therefore, outside of KVL application? The specified DSN contains an architecture mismatch Error Show that the set of all polynomials whose constant term is a multiple of 3 is an ideal of the ring $Z[x]$.   Attempt: Suppose then we need to show   if $p(x), q(x) \in I$ then $p(x) + q(x) \in I$  and if $r(x) \in Z[x]$ , then $p(x)r(x) \in I$ and $r(x)p(x) \in I$.  And $0 \in Z$, then $0 \in I$.  Then let   $I =$ {$a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + a_0$ | $a_i \in Z$, $a_0\in 3Z$}  $p(x)  = a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a$  and $q(x) = b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b$  Here we do not assume that $p(x) $ and $q(x)$ have the same degree.  Then $p(x) + q(x)  = [a_nx^n + a_{n-1}x^{n-1} + ...+a_1x + 3a]  +  [b_mx^n + b_{m-1}x^{n-1} + ...+b_1x + 3b] = 3(a + b) + (a_1 + b_1)x + ....+ (a_n + b_m)x^n$.  so $p(x) + q(x) \in I$  And $0 \in Z$ is also in $I$, since for $p(x) = 0x^n + ....+ 3(0) = 0$.  And let $r(x)= c_nx^n + ...+3c  \in Z[x] $.  Then $r(x)p(x) = 3(3ac) + (3b_1 + 3a_1) + ....+ a_nb_mx^{2n}$  So $r(x)p(x) \in I$.  Can someone please help? I am not really sure how to prove this. I am getting confused with the underscrips Any help or better approach would really help. Thanks in advance!