$$\frac{\sin(x)}{x} = 1$$ and $$\frac{1-\cos(x)}{x} = 0$$  I can't come up with any solution. Could you please help me?  Also: $$\lim_{x\to 0} \frac{1-\cos(x)}{\tan(x)\sin(x)}$$
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.
Given that $f:[0,1]\to \Bbb{R}$ is a continuous function, I need to show that $$\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)=e^{\int_0^1f(x)dx}$$  Writing $$y=\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$$ and taking the logarithm of both sides and interchanging limit and logarithm(since $f$ is continuous), I get $$\log y = \lim_{n\to \infty}\sum_{k=1}^{n}\log\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$$ Now I know that the limit of the Riemann sum $\lim_{n\to \infty}\sum_{k=1}^{n}\frac{b-a}{n}f\left(a+\frac{(b-a)}{n}k\right)=\int_a^bf(x)dx$. However if I want it to read $\int_0^1f(x)dx$ in the given problem, I would need the term inside the summation to read $\frac{1}{n}f\left(\frac{k}{n}\right)$. The way to do this it would seem is to take the series expansion of $\log(1+x)$ and ignoring the $x^2$ and higher order terms. Is it justifiable to do so? What is the easiest way to highlight the difference between two strings in PHP?  I'm thinking along the lines of the Stack Overflow edit history page, where new text is in green and removed text is in red. If there are any pre-written functions or classes available, that would be ideal. Intuitive idea of the Lipschitz function I'm trying to understand intuitively the notion of Lipschitz function.   I can't understand why bounded function doesn't imply Lipschitz function.   I need a counterexample or an intuitive idea to clarify my notion of Lipschitz function.  I need help  Thanks a lot Binding single key (such as 'F1') to multiple keys (such as 'Ctrl+C') in Gnome I know how to map a single key to a command using the gconf settings. I also know how to map a single key to another key (F1 -> ctrl) using Xmodmap. My question is how can I map a single key such as F1 to a multi-key operation such at ctrl+c. I am basically trying to set up a single key to simulate copy. I can think of two ways to possibly do this:   Map single key to multiple keys so that when the single key is pressed it is essentially pressing ctrl+c. Use gconf settings to execute the copy “command” upon the single key press. I am not sure if there is a “command” to copy text. Well, this question has been puzzling me for kinda long time, many people believe that orbiting astronauts feel weightless because they are "beyond the pull of Earth's gravity"...How far from the Earth would a spacecraft have to travel to be truly beyond the Earth's gravitational influence? If a spacecraft were really unaffected by Earth's gravity would it remain n orbit? If so, what is the real reason for weightlessness in orbit? How do I run .sh scripts? How to illuminate the darkness inside glass objects in Cycles? My model has a clear plastic (or glass) dome with objects inside.  The objects appear dark.  I've tried the glass shader with color 100% RGB, with very small IOR, and using just a refraction shader.    I know Cycles has problems with caustics, but I just want broad illumination to fall clean through the glass.  The light-to-column path can ignore refraction, but is blocked or tremendously attenuated. The camera-column path needs refraction, and seems to work fine anyway.  This demo image offers all optical paths of interest:   light upon object, seen directly light upon object, seen through glass (looks okay) light through glass onto object, seen directly. This is the shadow on the lit side of the column just below the glass tubing.  I expect full illumination, but there is shadow.  There is some light, but it's diffuse light from the floor. light through glass onto object, seen through glass. This     I did at one point try non-progressive rendering, and find extremely coarse light where there shouldn't be shadow, but this is not a practical way to make nice images. I've even tried cranking up certain sampling settings to many thousands, but still get bad noise though less, at the cost of long render times. I  would like to finish my creations sometime before "Star Trek, the Generation After the Next" becomes real life, if you know what I mean.  The next image is an example. I think Transmission in Sampling was 5000, but not sure.    I'm using Cycles in Blender 2.67 on a quad core Linux machine, 16GB RAM. Machine limits have never been a problem for 3D rendering.  Possibly, I screwed up some setting when I made my startup file, and just need to fix it. OTOH maybe this is just a limitation of path tracing, or of the way Cycles does it. If that's the case, what to real-world Blender artists do when they need to produce images on a budget and schedule?  What is the practical real-world fix? Php - regular expression to check if the string has chinese chars Automatically adding user and password to openvpn script I have a working script, which allows me to do a vpn connection. It looks like this:  openvpn --config VPNBook.com-OpenVPN-Euro1/vpnbook-euro1-tcp443.ovpn   All is good, but I have to input the publicly available vpn and password every time I run this script. (Don't worry, I'm not going to do banking or anything serious over a free VPN, this is more for learning).  This is the username and the password available publicly on their website:  Username: vpnbook  Password: he2qv5h   After firing up, the script prompts me for username and password which now I need to copy and paste.   Tue Aug 15 11:54:51 2017 library versions: OpenSSL 1.0.2l  25 May 2017, LZO 2.08 Enter Auth Username: vpnbook Enter Auth Password:    Is there a way of doing it so the script inputs the values on its own?  I tried echoing it, but it isn't that easy.  Or do I have to edit the command itself, somewhere in the files? If yes, can someone point me in the right direction for doing it? Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$ Alright so I've seen some postings about loop cuts and done some research. I am very new to blender and modeling overall. I am trying to loop cut this concrete barrier. Now I'm following a tutorial and kind of did my own thing in some areas. I cleaned it up following some instructions in another answer here on stack exchange.  I can't for the life of me figure out why I can loop cut around the whole bottom:   ...but not the whole middle:   I see this marked as answered by another question. That question refers to ngons and inside faces. Now once again I'm totally new to this. I looked on the inside. I don't believe I have inside faces or any ngons by the terms I've read online in my research.  Here is a wireframe picture and the inside view:     Maybe I'm missing something the answers on other posts are saying. If so and you know what it is I'd be very appreciative of what it is. How can I change the default program installation directory in Windows 7? Windows 7 is installed on my C drive, which is quite small.  I am very tired of instructing new programs to put their files on my larger D drive during installation; I would like to change the default drive.   says that you can use a registry hack, but I am giving Microsoft the benefit of the doubt and naively assuming that a configuration option exists somewhere.  It's 2010... do I really have to hack my registry to make a simple tweak like this?  Also, there's a  that explains how to move the "Users" directory and create a symlink, which could also work.  However, at the moment I have some apps in C:\Program Files, some apps in C:\Program Files (x86), and some apps in the corresponding folders on D:\, so it would be a hassle.  Also, my small OS boot drive is a 10k RPM WD Raptor, and I feel like that probably gives a speed boost to apps installed on it that need to read &amp; write to their directories a bunch.  I wonder if it actually matters. What is the definition of layer in neural network? How to effectively close a page? 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. I have a list with 15 numbers in, and I need to write some code that produces all 32,768 combinations of those numbers.   I've found  (by Googling) that apparently does what I'm looking for, but I found the code fairly opaque and am wary of using it. Plus I have a feeling there must be a more elegant solution.  The only thing that occurs to me would be to just loop through the decimal integers 1–32768 and convert those to binary, and use the binary representation as a filter to pick out the appropriate numbers.   Does anyone know of a better way? Using map(), maybe? Do universities hire graduates from lesser universities Recently, someone analyzed  and found that over half of the professors at the top 51 universities graduated from a top 10 university.  have also brought this up. From my personal observations, most schools do hire graduates from better ranked schools.  Is this because of the competitive job market? We have so many good applicants, we have to narrow it down some how! Or is it simply that these schools produce the most PhDs? Has this always been the case? How rare are exceptions to this? I know of a few people who graduated from a top 75 school and got hired at a top 50 school. But what about bigger gaps? The top 10 schools seem to just swap graduates, do they ever hire from a 50+ ranked school?  Update 2018: I have accepted a tenure-track position at a top 75 department at an R1 university immediately after graduating from an unranked department at an R2 university. It does happen! This may or may not generalize to other fields and countries. Why are so many forces explainable using inverse squares when space is three dimensional? Do we have to tune the number of trees in a random forest? Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second.   When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards the higher numbers. I'm being asked to "describe" the distribution. I know that we can look at each distinct outcome, like:  max(1,1) = 1  max(1,2) = 2  max(1,3) = 3  max(1,4) = 4  max(1,5) = 5  max(1,6) = 6  max(2,1) = 2  max(2,2) = 2  max(2,3) = 3  max(2,4) = 4  max(2,5) = 5  max(2,6) = 6  max(3,1) = 3  max(3,2) = 3  max(3,3) = 3  max(3,4) = 4  max(3,5) = 5  max(3,6) = 6  max(4,1) = 4  max(4,2) = 4  max(4,3) = 4  max(4,4) = 4  max(4,5) = 5  max(4,6) = 6  max(5,1) = 5  max(5,2) = 5  max(5,3) = 5  max(5,4) = 5  max(5,5) = 5  max(5,6) = 6  max(6,1) = 6  max(6,2) = 6  max(6,3) = 6  max(6,4) = 6  max(6,5) = 6  max(6,6) = 6  But is that the most effective way to look at a distribution like this? Is there a faster and more concise way to describe this distribution other than saying the probability of each outcome?  i.e.,  P(max = 1) = 1/36  P(max = 2) = 3/36 = 1/12  P(max = 3) = 5/36  P(max = 4) = 7/36  P(max = 5) = 9/36 = 1/4  P(max = 6) = 11/36  So, obviously, I understand how to LOOK at a discrete distribution and see what the probabilities are, but not until I look at each possible outcome. But what does it mean to "describe" a distribution? This isn't uniform or any sort of discrete distribution we're taught. Can someone provide clarity? Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$?  If so, is it true that $\ell=\lim_{x\to a}f(x)\le f(x) \ \ \forall x\in (a,b)$? Please provide proofs.