I'm wondering if it's possible to do something like this in Latex?    I have seen the similar question , but I'm wondering if there is a way to shade, have dotted lines, and have the dot? Ideally, I'd want to have a way to show how different $\epsilon$ values affect the $\delta$ values, i.e., as $\epsilon$ gets smaller, there is a $\delta$ that works.  So far, I've simply recreated the image in PowerPoint which I then copy paste in my tex document.
I'm trying to do something similar to this epsilon-delta graph I've found.  Specifically, I'd like to copy the placements and arrows, like the L+e and L-e coming out of L. Same thing with the x0 + d and x0-d coming out of the x0. Can someone help me with this? This is what I have so far.    \begin{tikzpicture} \begin{axis}[ ,         legend pos=north east,         axis x line=middle,         axis y line=middle,         x tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         y tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         grid = major,         width=8cm,           height=6cm,         grid style={dashed, gray!30},         xmin=0   ,     % start the diagram at this x-coordinate         xmax= 4,    % end   the diagram at this x-coordinate         ymin= 0,     % start the diagram at this y-coordinate         ymax= 11,   % end   the diagram at this y-coordinate %axis background/.style={fill=white},         xlabel=$x$,         ylabel=$f(x)$,         tick align=outside,         enlargelimits=false] % plot the stirling-formulae %\addplot[domain=0:200, red, ultra thick,samples=100] {K/(1 +    ((K-X0)/X0) * exp(r*x))    }; \addplot[domain=0:200, red, ultra thick,samples=100] {3*x-1 };  \addplot[line width=1pt, dashed, mark=none, black] coordinates {(0,8) (3,8)}; \addplot[line width=1pt, dashed, mark=none, black] coordinates {(3,0) (3,8)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(3.33,0) (3.333,9)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (2.666,0)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (0,7)}; \addplot[line width=1pt, solid, mark=none, blue] coordinates {(3.333,9) (0,9)}; \addplot[holdot] coordinates{(60,8.66)};  % 1\addlegendentry{$f(x) = 10 \sin (x)$}  \end{axis}   \end{tikzpicture}   which gives
Open an app on startup/after booting I need to force a selected application to start up each time my Android-based device is (re)started.  Is there anything I can use (configuration setting, external application etc.) to accomplish this task? Or is this kind of behavior limited to home screen managers only? Value of $\frac{x^n -a^n}{ x-a}$? I'm trying to recover a Windows system from backups and can't seem to find where Chrome saves its bookmarks.   Does anybody know where Chrome saves its bookmarks on Windows? Should I postpone my trip to Paris in 2nd week of Jan in light of recent protests? I’ve a date and I really waited long to make this happen. My gifrfriend is a bit worried about the security issue in Paris. Which locals think that the media is exagerrating the issue and the issue is small and non-threatening for tourists? What spell targets count for the Graviturgy Magic wizard's Gravity Well feature? The  feature from the  (EGtW, p. 185) triggers &quot;whenever you cast a spell on a creature&quot;, and has an effect &quot;if the target is willing to move, the spell hits it with an attack, or it fails a saving throw against the spell.&quot; It's not clear to me what kinds of spells can actually trigger this feature. Based on the singular phrasing of &quot;on a creature&quot; and the later use of the word &quot;target&quot;, I would normally read this as only applying to spells that explicitly target creatures (and perhaps only those with a single target, in contrast to the phrasing of the  feature of the  wizard, for example). However, spells that target a location can use the word target in a different sense (see ). What's actually going on here? Does this feature only work on targeted spells? Or does it function for area of effect spells as well? Or is this just an ambiguous phrasing that will rely on the DM's ruling? Rotating a gyroscope's output values into an earth-relative reference frame For whatever reason, I seem to have gotten myself twisted into quite a confusion on how to process data from my 3 axis gyro and 3 axis accelerometer to get gyro rotation values related to the earth's reference frame. i.e. Given my accelerometer measures gravity (we assume it's stable) along the sensors (x,y,z) reference frame, how do I adjust the values of my gyro, measured along the sensors same (x,y,z) reference frame such that I am actually measuring the rotating about the Earth's frame x (North), y (West), and z (downwards) direction?  I just can't seem to wrap my head around the use of the rotation matrix. Do I simply employ a 3d rotation matrix, and multiply the vector, \$[gyro_x, gyro_y, gyro_z]^T\$, by a rotation matrix, R? I then assume the rotation angles, are somehow derived from the filtered accelerometer outputs, but I'm having trouble deciding on what those should be.  I have yet to find a concise reference on how to move the rotation and acceleration values from the sensor reference frame into the earth global reference frame. Perhaps someone can help? Thanks When is a clause not essential? I am having trouble figuring out when to use commas to set off &quot;nonessential&quot; information.  Sometimes it's obvious:  Bob, who is thirty years old, is an alcoholic.  But other times I'm not sure:  The day he quits drinking * he will start a llama farm He has his heart set on owning El Duderino ranch * in New Mexico.  In the first case, my ear says that there should be a comma at *, even though information before it seems essential to me.  In the second case, the stuff after * is not essential, and yet it seems a little much to use a comma. Even that last sentence I wrote confuses me.  &quot;In the second case&quot; seems essential but I used a comma.  Is this correct? Does it depend on personal style and the length of the clause?  Or perhaps I'm misinterpreting the meaning of &quot;essential&quot; in this context? Why no tag Badges for Good questions Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". MacBook Pro - FaceTime HD Camera - Hardware "Failure'? System Report>Camera>No video capture devices were found     USB>Device Tree> No FaceTime HD Camera (inbuilt)listed    What can be the cause for "camera hardware failure"? Could faulty iSight cam wifi antenna cable linkage between logic board and camera board (in display)result in Camera hardware not "found"? Neopronouns or trolling? It’s the  question: Am I looking at someone’s legitimate pronouns or trolling?   (It’s unfortunate I have to ask, but this is just where we are right now.)  I’ve seen “my pronouns are [insert ridiculous fake stuff here]” trolling recently (in posts) when before I never did. Also stuff that looks to be trolling, but could also maybe not be. Trolls deserve red flags while everyone else specifying pronouns does not, right?  (To clarify, the problem is not just about not wanting to use a troll's words. The purpose of some of this trolling is to try and convince people that transgender people—or their neopronouns—are stupid or unreasonable. Ignoring the trolls allows us to avoid the first problem, but not the second since it allows them to still reach people with their message, which is not something that I want to see promoted.)  Some neopronouns are clearly legitimate. Some trolling is clearly trolling. But unfortunately there’s a big grey area where it’s not clear to me what is what.   Does anyone have a good method for dealing with this? There just doesn’t even seem to be a good way to ask about someone’s specified pronouns in such a way that doesn’t sound insulting if the user isn’t trolling.     Note: A while ago, I did some research on neo-/third person pronouns which can be found . There are several important things I found:   There are more neopronouns than I could ever hope to keep track of The most popular neopronouns are very uncommon. On the other end of the spectrum, it looks like some neopronouns are neologisms used by a single person  Some neopronouns come from unlikely places, such as SciFi I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$. How to group distinct values from a list using odata rest api Hi I have a list with two fields "Regions" &amp; "Countries" I am using a rest api to retrieve the list of just regions like so: http:mysitename/_api/web/lists/getByTitle('TaxTool')/items?$select=Region  In the results there are lots of duplicates and I was wondering how to return unique values. Why doesn't the Doctor go back to Gallifrey before the Time War? According to , it’s just the Time War that is time locked. So why does Eccelston's Doctor tell Rose:     My whole planet is gone. My people, my family, all gone. Don't you think I want to go back and save them?   Why hasn't he ever gone back to before the war?   All the other Doctors basically see Gallifrey as gone from them, and when you say the Doctor cannot go back on his own timeline, there will have been plenty of time periods he could go to; maybe to a time before they could regenerate, or maybe when Time Lords were using Tardises — as in "The Doctor's Wife" she says to Matt Smith:     I was already a museum piece when I stole you Fingerprint GUI not installing in Ubuntu 20.04 I'm trying to install fingerprint-gui on my Lenovo Ideapad 520 having Ubuntu 20.04 on which fingerprint isn't working for now. I tried going directly to Settings → Users → Automatic login and enabled the fingerprint login, but the error is:  Could not access any fingerprint readers Please contact your system administrator for help!  I also tried to install Fingerprint GUI: sudo apt-add-repository ppa:fingerprint/fingerprint-gui &amp;&amp; sudo apt-get update  sudo apt-get install libbsapi policykit-1-fingerprint-gui fingerprint-gui  but encountered this error: E: Package 'policykit-1-fingerprint-gui' has no installation candidate E: Package 'fingerprint-gui' has no installation candidate  What could be the problem here? Can matter really fall through an event horizon? Proper fstab entry to mount a samba share on boot? Pointwise convergence of $h_n(x) = x^{1+\frac{1}{2n-1}}$ Okay, maybe I'm missing something obvious but consider $h_n(x) = x^{ 1+ \frac{1}{2n-1}}$ on the set $[-1,1]$. Now consider a fixed $x \in [-1,1]$, we have:  $$\lim_{n\rightarrow \infty} h_n(x) = x \lim_{n \rightarrow \infty} x^\frac{1}{2n-1} = x$$  But my book says it's $|x|$? Why? Punctuating a quoted question within a quoted statement I'm working on a story, and I find myself faced with something of a quandary.  Specifically, I need to know the proper style for the following:     “I have thousands of them,” the commissioner answered, “but I’ll start with ‘What do you need?’.”   Is it proper form to have both the question mark and period, or should one of them be dropped? Does anyone know if there is an API to get the current monitor state (on or off) in Windows (XP/Vista/2000/2003)?  All of my searches seem to indicate there is no real way of doing this.    tries to use  which according to Microsoft's docs does not work for display devices.   In Vista I can listen to  but I do not seem to get events when the monitor is turned off manually.   In XP I can hook into WM_SYSCOMMAND SC_MONITORPOWER, looking for status 2. This only works for situations where the system triggers the power off.   The WMI Win32_DesktopMonitor class does not seem to help out as well.   Edit: Here is  on comp.os.ms-windows.programmer.win32 indicating there is no reliable way of doing this.   Anyone else have any other ideas? Dual boot windows 8 and Ubuntu with Windows 8 Boot manager I have two partitions on my hard-didk , I have installed ubuntu on my 1st partition and windows 8 later on another partition.Now i can only boot into windows 8 because it doesn't recognize Ubuntu.  How would i dual boot my PC without using grub . I would like using Windows 8 boot manager as its pretty neat.  This is what i have tried :  I tried using easyBCD but it doesn't work.It causes the boot manager to switch to windows 7 Boot Manager .  EasyBCD creates a C:/NST/AutoNeoGrub0.mbr file .  As EasyBCD was causing the boot manager to switch i created a manual entry to the BCD store uisng windows bcdedit command and point it to the C:/NST/AutoNeoGrub0.mbr file created by EasyBCD. This gave me the Windows 8 Boot-Manager with Ubuntu Entry,but grub is unable to find the root partition.  Is there any other work around or solution for this ?  Any help would be appreciated.  Note: The windows 8 boot Manager is sky blue color interactive menu with mouse and other options and windows 7 boot manager is the normal black and white one where you can only use your keyboard  I don't want to install grub because it would prevent my windows 8 from updating as its a genuine MSDN version and various other reasons