For complex number $z$, how to prove the limit exits by definition  $$\lim_{z\to 0} \frac{\sin z}{z} =1$$  By def I've tried, but I got a difference always greater than 1..
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.
My Command is below and its print Free CPU and Used CPU below format.  I want to restrict awk output to 2 decimal place.  grep 'cpu ' /proc/stat | awk '{usage=($2+$4)*100/($2+$4+$5)} {free=($5)*100/($2+$3+$4+$5)} END {print "Used Cpu:",usage "%"} {print "Free Cpu:",free "%"}'   Output :   Free Cpu: 98.9588%  Used Cpu: 1.04125%   Desired output :  Free Cpu: 98.95%  Used Cpu: 1.04% How to hide "internal" polygon boundaries? Print lines between (and excluding) two patterns Which is the recommended way to plot: matplotlib or pylab? Conditionals in English are usually formed by using if with normal word order; but for the three past (subjunctive) forms were, should, and had, it is also possible to express the conditional through subject–auxiliary inversion alone, with no if in the conditional clause.  Does forming conditionals in this manner differ semantically from if-conditionals? Is some aspect of the conditional statement or the conditionality emphasised more in one version than in the other? Or are there differences in how and when they are used? Or are they simply completely interchangeable?  For example:     If I were you, I would try it again   Were I you, I would try it again      If I had seen it, I would have told you   Had I seen it, I would have told you      If you should drink, don’t get behind the wheel   Should you drink, don’t get behind the wheel What is the difference between a framework and a library? What is the difference between a framework and a library?   I always thought of a library as a set of objects and functions that focuses on solving a particular problem or a specific area of application development (i.e. database access); and a framework on the other hand as a collection of libraries centered on a particular methodology (i.e. MVC) and which covers all areas of application development. Computing matrices of linear transformation under different basis Came across a problem in my textbook and I am having trouble with it, the book provides an answer but no workings. Looking to get an explanation of what is going on  help I prepare for an exam. If we let $T : \mathbb R^2 \to \mathbb R^2$ be a linear transformation. Consdider two bases $V$ and $W$ of the vector space $\mathbb R^2$, where $V=\{v_1,v_2\}$ with $ v_1 =  \begin{bmatrix} -2 \\1   \end{bmatrix}$ and  $ v_2 =  \begin{bmatrix} 1 \\1   \end{bmatrix}$ and where $W = \{w_1,w_2\}$ with  $ w_1 =  \begin{bmatrix} -1 \\4   \end{bmatrix}$ and  $ w_2 =  \begin{bmatrix} 0 \\1   \end{bmatrix}$. Suppose that  $ A =  \begin{bmatrix} 1&amp;2   \\-1 &amp;4    \end{bmatrix}$ is the matrix of $T$ under basis $W$. I need to find the matrix $B$ the matrix of $T$ under basis $V$.  The solution is:   $ B = P^{-1}AP$ = $   \begin{bmatrix} -30&amp;22   \\-48 &amp; 35    \end{bmatrix} $  But I don't know why. What visa does my wife need to travel to the UK with me, if I'm going there on a business visa? I am resident of India and just got UK Business visa ( 6 months, multiple entry) .. I am going for a week on business trip. I want to take my wife along with me and at the end of trip.. I want to extend my trip for 3 days and we want to spend some time on leisure.   What visa I should apply for my wife.. visitor visa? should she mention in the visa form that she is going along with me and staying with me while I am on work and later we are spending time on leisure.   While filling my business visa form, there was a field asking if anyone else is going with me.. I selected no.. as there were no plans.  Also is it ok to spend time on leisure on UK Business visa Please help identify this 2-part TV movie: Alien woman crash-lands on earth, falls in love, 199x How can I recover photo files from a memory card with a corrupt filesystem? I know this is not specifically a photography issue but since many of us with prosumer/professional-level DLSRs use Compact Flash cards, it is bound to happen to others as well.  I have a Sandisk 16GB Ultra card that occasionally had the camera reporting inability to access the card. I had to reinsert the card to continue using it. I was able to read the card and transfer the photo files to my computer drive. Today, the same thing happened again, just that nearing the capacity of the card, it failed one last time and the camera was subsequently unable to read it anymore. Despite reinsertion.  My other CF cards never once exhibited this problem, so it's pretty much isolated to this card.  The card is similarly unrecognisable on my computer CF card reader. I think the filesystem is toast, but I suspect the photo files are still intact. I would like to find ways where I can inspect the raw binary data in the card and hopefully recover the photos? Why are papers without code but with results accepted? I just started reading some papers (Computer Science, specifically Computer Vision) and thought "now let's look at the source code" and was quite astonished that most of the papers don't have any source code implementing the described methods to look at, while claiming some performance or being better than other papers.  How do these papers get accepted in journals / conferences? Do people have to submit their source code privately to the reviewers at least, so that they can reproduce the experiment if possible.  Do most journals / conferences just "trust" that people who submit the paper really implemented the theory and got those exact results?  I always had this idea that any experiment should be reproducible by others else it's not scientific justified. Been wondering about this the last few days. Command to move a file to Trash via Terminal I have found film reels some with tails of film others not. Is there a way of telling if they have been exposed/used in a camera? Film types are for 35mm and one Kodak Advantix. Calculating roundness/compactness of polygon? I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?) When is $2^n -7$ a perfect square? This came up while solving another ENT problem. I want to ask when is: $$2^n -7 \text{ where } n\geq 3$$ a perfect square? Specifically, I also wanted to know what would be the solutions when $n$ is odd? How should I solve this?  I can check that $n=3, 4, 7$ are solutions but cannot find more. As it is of the form $4k+1$, it doesn't help as well. My current understanding:  $\vdash$ is "syntactic consequence", e.g. if we have $a \vdash b$ it means if we've written down $a$ as something we know, we can immediately write down $b$ as something we know. This is saying nothing about whether any of these pieces are true, false, something else, etc, it's just saying that we can say something about $b$ by virtue of being able to say something about $a$.  $\vDash$ is "semantic consequence", e.g. if we have $a \vDash b$, then it (usually) means if $a$ is always true in every interpretation (i.e. no matter how we mess with any of the underlying variables driving $a$'s valuation), then $b$ is true as well. However I don't know if this is necessarily just a "true"-only thing. It seems conceivable that we could make one for falsehoods too, i.e. if $a \vDash b$ might mean if $a$ is always false then $b$ is always false, of if $a$ is always "dog" then $b$ is always "dog", etc. But I am unsure of this one.  $\to$ is material implication, which exists solely within the logic system (as opposed to the previous two metalogical relations) and has a specific semantic meaning / definition in terms of a truth table in something like prepositional logic ($a \to b$ is always true unless $a$ is true and $b$ is false). It's defined this way simply because it is a helpful / useful model to describe things like "whenever $a$ is true, $b$ is true".  Do I have the right idea so far? I was having a discussion with a friend of mine about some normal group properties and then came up with the question "if G is not commutative, then is there always a subgroup that is not a normal subgroup?" It's probably more easy to solve this in the following form:$$\forall H \leq G : H  \lhd G \Rightarrow \forall a,b \in G : ab=ba$$ My question is, can anybody give a proof, or a counter-example (because I don't think it holds) of this theorem? Thanks! I have some Python code that makes Raster Attribute Tables for GeoTiffs using GDAL.    In ArcGIS I can see these attribute tables fine and they behave how I like (as classifications) but in QGIS the whole concept of an attribute table seems to be missing.    Is there any way to access a RAT in QGIS? Why are "south" and "southern" pronounced with different vowels? I was wondering why we pronounce the vowels in the words south and southern. They seem to be very closely related to each other. Both refer to the same direction. South is a noun and Southern is an adjective, which I don't think caused them to have a different vowel. Oxford Dictionary shows that south is /saʊθ/ and southern is /ˈsʌð(ə)n/. In short, south has a long vowel and southern has short. Is there a simple way in Perl that will allow me to determine if a given variable is numeric?  Something along the lines of:  if (is_number($x)) { ... }   would be ideal.  A technique that won't throw warnings when the -w switch is being used is certainly preferred.