It is well known that the function $f(x)=1/x$ is not defined for $x=0$. However, simply multiplying $f$ by the function $g(x)=x$ gives a constant, very well defined, function, even at $x=0$. How can it be that multiplying by $g(0)=0$ something which doesnt exists ($f(0)$) can actually yield a result ?  Moreover, if I multiply the whole $f(x)$ by $0$ it obviously gives me $0$ at $x=0$ where i had $1$ in the previous case. This seems to indicate that either :   there is something more to the function $f$ at $0$ than not being defined there is something wrong with the way I think about the product of two functions.
My question is whether $x/x$ is always equal to $1$. I am mostly intersted in real numbers and particularly wonder whether $x/x$ is defined at $x=0$.  On one hand the division should simplify to $1$, on the other hand you should not be allowed to divide by zero.   I have been trying to find whether the simplification 'goes first' or whether the division causes trouble first, but it has proven impossible for me to find usefull search terms.     Note that this question arose after reading  and its first comment.
Single quote within double quotes and the Bash reference manual In section 3.1.2.3  titled Double Quotes, the Bash manual says:      Enclosing characters in double quotes (‘"’) preserves the literal   value of all characters within the quotes, with the exception of ‘$’,   ‘`’, ‘\’, and, when history expansion is enabled, ‘!’.   At the moment I am concerned with the single quote(').  It's special meaning, described in the preceding section, section 3.1.2.2 is:     Enclosing characters in single quotes (') preserves the literal   value of each character within the quotes. A single quote may not   occur between single quotes, even when preceded by a backslash.   Combining the two expositions,    echo "'$a'"   where variable a is not defined (hence $a = null string), should print $a on the screen, as '', having it's special meaning inside, would shield $ from the special interpretation. Instead, it prints ''. Why so? addtocontents not working with certain other packages Relationship between $R^2$ and correlation coefficient Travelling with two different passports The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ How to compute this limit:  $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$  Please give me some hint. I’m able to copy-paste from any application to any application. I’m able to copy from a comment or from a question or from an answer. I’m able to paste to a question or answer. However, if I attempt to paste into a comment, nothing happens. I am able to subsequently paste into another application, confirming that the text is actually on the clipboard.   However, copying SE links will paste them successfully, but all other text - including other links - still fails. This leads me to believe that it’s a SE issue and not an Apple one.   This has been an issue since 11.0 beta and persists through iOS 13 release. Before asking my question, I'll use an example to clarify the idea.  Say that a company used to sell horses and then they turn into a sheeps business. While some of the old pages can be mapped to the new pages using a 301 redirect, because they're really related in the content and the company don't want to lose users and link juice, there are a lot of pages strictly related to horses that they want to shutdown.  What's the best practice? Is there a way to say to users and Search Engines the content you were looking for won't be available any more?  Basically a 301 redirect of the closed pages to new ones seems not so good, because you'll show a content totally unrelated to the searched one.  P.S: I'm aware that probably the best fallback is to simply map those pages to the homepage but I hope in a smarter solution. In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? At school I learnt the  to remember the resulting direction of different phenomena, such as geometrical cross products, mechanical torque, or the direction a screw will move when turning it. This directions are however decided by humans, and we could have easily defined the directions in the opposite way.  However the right hand rule also predicts phenomena which is not arbitrarily set by humans, but is intrinsic to the nature. If you have a coil in which you put some current, and the nature creates a magnetic field in a direction perpendicular to it, the direction depends on the right hand rule.  My question is, why this direction and not the opposite? Nature tends to be symmetric and not have apparent preferences for two equal options, so in a so why is one direction preferred over the other one? Why doesn't a "left hand" rule apply? Extrude and bevel an imported svg curve I exported an svg 1.1 file from Adobe Illustrator CS6. I joined all the shapes by pressing Ctrl+J. I set the curve shape to 2D. Then I extruded using the right hand side panel:    So far so good. But when I try to apply even a slight bevel, all hell breaks loose:    It does get less insane if I bring the bevel down by multiple factors of 10... But then I really don't have much of a bevel.  Am I doing something wrong? Am I using the wrong kind of object? Should I be doing this with meshes? Do my curves just need cleaning up? Moving an already-running process to Screen My office job routinely sees me connected to a Linux box via VNC.  Sometimes I start a remote job on the console, and realize later that it runs much longer than expected. (Should have started that one under Screen in the first place...)  I don't want to keep my workstation running overnight just to keep the VNC session open; I would like  to move that already-running remote job into a Screen session (on the remote box), so I can power down the workstation (and reconnect next morning).  How can this be done, if at all? What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? How to add D-link dwm-156 to usb switch mode for automatic eject of driver CD / switch to modem mode? I am running Ubuntu 13.10, 32-bit.  I had every time need to go terminal to "eject" the windows driver CD,nor gone to disk &amp; eject windows driver CD to make the d-link dwm-156 gsm modem functioning through network manager.  It seems that usb mode switch is not working.  But on my friend's computer with ubuntu 14.04 ,usb mode switch is working and i can connect d-link dwm-156 to mobile broadband through network manager as soon as i get in.  Is there any way to make usb mode switch to eject the windows driver CD of d-link dwm-156 in my ubuntu 13.10,32-bit at start up?   The lsusb output is Bus 002 Device 013: ID 2001:a706 D-Link Corp before ejecting the CD (modem as storage device of windows driver).  The lsusb output is Bus 002 Device 005: ID 2001:7d01 D-Link Corp after ejecting the CD (as modem).   I don't want to use the driver in the CD,as i need to send sms &amp; find the data usage. Is ¬¬(¬¬P → P) provable in intuitionistic logic? What is the correct name for a "product function" on a monoid? Where Can I Give Feedback on Ubuntu? I'm running Ubuntu 11.04 Beta 2. Is there a way, or a place, to give feedback on this system? Not file bugs, just give feedback. The table cards were A,K,K,8,3. All four suits were on the board. Player one had A,8. Player two had A,3.   Is this a split pot? Surface of genus $g$ does not retract to circle (Hatcher exercise) I'm trying exercise 9 on page 53 in  but I need some help with it. The exercise is:  In the surface $M_g$ of genus $g$, let $C$ be a circle that separates $M_g$ into two compact subsurfaces $M_h^\prime $ and $M_k^\prime$ obtained from the closed surfaces $M_h$ and $M_k$ by deleting an open disk from each. Show that $M_h^\prime$ does not retract onto its boundary circle $C$, and hence $M_g$ does not retract onto $C$. [Hint: abelianize $\pi_1$.] But show that $M_g$ does retract onto the nonseparating circle $C^\prime$ in the figure.    My first question is: assume there wasn't the hint, how would I think of abelianising? What does it mean exactly?  I thought I could do this by contradiction: if it retracts the induced map $i_\ast : \pi_1(C) \rightarrow \pi_1(M_h^\prime) $ is injective. I know $\pi_1(C) \cong \mathbb{Z}$, then I computed $\pi_1(M_h^\prime) \cong \mathbb{Z}$ and then I think I'm stuck. Right? Do you agree with $\pi_1(M_h^\prime) \cong \mathbb{Z}$ and being stuck after that?  What do I need to know to make progress? Many thanks for your help! How to merge vertices in Blender? I am attempting to create my first 3D character just for practice. Being one of my first 3d models I going through the learning curve needless to say. In doing so I find myself creating many extra vertices. Where we will say 10 vertices on a rounded shape are spaced out on the equator and on the other north pole end of the edges they are very close. The close end could all be one vertex. Is it possible to say merge all points to one point and delete the rest? How is Selenium RC different from Selenium Web driver and some good pointers for a naive beginner to learn to scripting please? Thanks! Implement MicroController Power Switch and re-use the button for other features This is similar to other questions, but I feel it is different because of the reuse of the "power" button.   I want a "latching" Momentary Power button.  Behavior:   User Pushes momentary button, holds down as long as necessary and the device powers up. Between Power On and Off the same button will be re-used for other functions  When it is held down for 5 +/- seconds the MicroController will turn itself off.    Design Constraints:   Battery provides between 3.7V to 2.9V during it's usable life The Voltage regulator requires Vin +/- 0.3V on the Enable Pin, 3.1V output. MicroController operates at 3.1V MicroController Output Pin High state is 2.7V Max   Debouncing: In this case I'm not concerned about debouncing the switch, if the user doesn't hold it down long enough for the MicroController to set it's Digital Output pin  High the device doesn't power on.  I've worked up the following schematic, which kind of works.    It assumes 3.7V from the battery. Uses an Analog Input for the Power button, which then uses the variation in voltage to determine if the button has been pressed. In the current design the difference is 0.12 V (from 3.4 to 3.52), but the MicroController has a 12 bit ADC so that shouldn't be a problem, in addition I can adjust the sensitivity range of the ADC.   Questions:  What is the difference between reality and the simulator?  Is there a better way?  How can I get a greater voltage difference on the button input? I've tried many different combinations but they increase the voltage range above the 3.6V Input High Maximum specified in the datasheet.  I'm not excited about the "Leakage current" when the uC Controlled Power Pin is Low, any suggestions?   Thank you for any ideas/suggestions/answers.