Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(c)}{x_k-c}=f'(c)$ where $\lim_{k\to \infty} x_k =c $ Question.  Suppose that a function $f$ is defined on an interval $I$, $c$ is a point of $I$, and $\{x_k\}$ is any sequence of points in $I$, no term of which is $c$, such that $\lim_{k\to \infty} x_k =c $. Define a sequence $\{y_k\}$ by $y_k=\frac{f(x_k)-f(c)}{x_k-c}$.  a)Prove that $f'(c)$ exists if and only if $\lim_{k\to \infty} y_k $ exists and has the same value for every such sequence $\{x_k\}$.  b)Prove that, if $f'(c)$ exists, then $\lim_{k\to \infty} y_k=f'(c) $ for every such sequence $\{x_k\}$.  My attempt(...?)  a) Let $\epsilon&gt;0$ be given.  Since $f'(c)$ exists, $\lim_{x\to c} \frac{f(x)-f(c)}{x-c} $ exists.  (...)  There exists $k_0\in \mathbb N$ such that for $k\ge k_0$, $\vert \frac{f(x_k)-f(c)}{x_k-c}-L \vert&lt; \epsilon$.  How can I find the link between $\frac{f(x)-f(c)}{x-c}$ and $\frac{f(x_k)-f(c)}{x_k-c}$?  It just seems so trivial to me and I cannot fill the blank.  Any advice would be appreciated!
Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$ Question.  Suppose that a function $f$ is defined on an interval $I$, $c$ is a point of $I$, and {$x_k$} and {$y_k$} are any two sequences in $I$ such that $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k&lt;c&lt;y_k$ for all $k$. Prove that, if $f'(c)$ exists, then $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   My Attempt.  Let $\epsilon_k=\frac{1}{k}$ be given.  For $\epsilon_1=1,$ there exists $\delta_1&gt;0$ such that $x_1,y_1\in I, x_1,y_1\in N'(c;\delta_1)$ implies $\vert \frac{f(x_1)-f(y_1)}{x_1-y_1} -f'(c) \vert&lt;\epsilon_1.$  For $\epsilon_2=\frac{1}{2},$ there exists $0&lt;\delta_2&lt;$min$\{\vert x_1-c \vert,\vert y_1-c \vert,\epsilon_2\}$ such that $x_2,y_2\in I, x_2,y_2\in N'(c;\delta_2)$ implies $\vert \frac{f(x_2)-f(y_2)}{x_2-y_2} -f'(c) \vert&lt;\epsilon_2.$  Similarly, for $\epsilon_{k}=\frac{1}{k},$ there exists $0&lt;\delta_k&lt;$min$\{\vert x_{k-1}-c \vert,\vert y_{k-1}-c \vert,\epsilon_k\}$ such that $x_k,y_k\in I, x_k,y_k\in N'(c;\delta_k)$ implies $\vert \frac{f(x_k)-f(y_k)}{x_k-y_k} -f'(c) \vert&lt;\epsilon_k.$  Then by taking the limit to infinity, we obtain $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   It really took me a lot of time to finish the proof and I need this to be checked by an expert:) Is the proof okay? Any response would be appreciated!
Bypass BIOS password set by faulty Toshiba firmware on Satellite A55-S1065 laptop? I have a doubt. I need help.  Can the basic axioms of equality be presented as "open axioms"?  I) (reflexivity) $\qquad x = x$              II)  (Substitutivity) $\qquad (x = y) \to \big(F (x, x) \to F (x, y)\big)$ Possible (but not serious) glitch in close/reopen I just had one of my questions closed. (Hey - it happens to everyone) When I clicked on reopen - lo and behold - it worked. I cast a reopen vote!  Is this a glitch? I know we need 4 more votes anyway but still...  And no  tag please :) Get the title instead of the number of a referenced chapter/section? How to serialize a JObject without the formatting? Invariance of a maximally entangled state under unitary operation $U \otimes U^\dagger$ Apparently the (d-dimensional) maximally entangled state, $|E \rangle = \sum_{i} |ii\rangle /\sqrt{d}$ is invariant under operations of the form $U \otimes U^\dagger$. I want to prove this result, which amounts to showing that  $$\left(\sum_i U | i \rangle \otimes V |i \rangle = \sum_i | ii \rangle \right)\Rightarrow (V = U^\dagger )$$  I have no idea how to even start. I suppose it's some simple linear algebra result, but I don't see it. A hint would be appreciated. Word for giving animal characteristics (esp. physical ones) to humans I am writing a paper about Art Spiegelman's Maus, specifically the metaphor that Spiegelman creates by depicting his obviously quite human characters with the heads of various animals, or a couple times animal masks. The metaphor is used at times to bestialize/animalize the acts of the Nazis and, in some cases, to show the arbitrary nature of the classes/groups/divisions they and others applied to human beings.  I considered "chremamorphism," but this describes the attribution of the characteristics of an object to, or the objectification of, a human. I also considered "animalize" and "bestialize," but those describe some of the effects that can be produced by Spiegelman's metaphor, not the metaphor itself - in many scenes in the book, the characters are extremely human, and I think that the negative or degrading connotations of these words makes them inappropriate for this use. Nor do "anthropomorphism" or "personification" fit, because Spiegelman is not applying human characteristics to animals but the other way around. None of the words that I can think of quite fit.  What is a word that accurately describes Art Spiegelman's depiction of obviously human characters as having animal body parts or wearing animal masks in the comic book Maus? C++ 11 Thread initialization with member functions compiling error This question is similar, but not identical to:   Whereas the aforementioned question is about whether the hand’s movement is measured from the hand’s point of view or from the caster’s, this question revolves around the total distance the hand can travel to reach its destination.  To illustrate, let me give an example: Two narrow stone pillars stand next to each other, 10 feet apart. They are several kilometers high and on top of each pillar stands a wizard. One wizard casts Maximillian's Earthen Grasp and lets the hand made from compacted soil rise from the ground on his own pillar. With another action, he wants to move the hand in order to restrain the other wizard, who is within range of 30 feet.  Would the hand retract into his own pillar, then rise again from the ground on the other pillar? Or would the hand move several kilometers down the first pillar, across the floor, then several kilometers up the second pillar? What if both pillars would float in the Astral Plane, without a direct ground-based connection between them?  This might seem like a stupid and unnecessary convoluted question, but lo and behold, something similar happened in one of my games. How to make an image center (vertically & horizontally) inside a bigger div I have a div 200 x 200 px. I want to place a 50 x 50 px image right in the middle of the div.   How can it be done?  I am able to get it centered horizontally by using text-align: center for the div. But vertical alignment is the issue.. If $AB = I$ then $BA = I$ If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix. I have a script myscript.sh  #!/bin/sh  echo $1 $2   which is used something like ...  ./myscript.sh foo bar   which gives me an output of ...  foo bar     but how can i make this script include custom command options? for example ...  ./myscript.sh -a foo and corespond to $1   and  ./myscript.sh -b bar and corespond to $2 Custom macro for multicolumn with & without xparse The following example has two normal LaTeX \newcommand macros and two xparse \NewDocumentCommand for specifying multicolumn text. The only difference between each pair of commands is one has the alignment character as part of the macro and one does not.  However, the xparse version that requires the alignment character to be outside of the macro does NOT work and results in Misplaced \omit. error.  \documentclass{article} \usepackage{xparse}  \newcommand{\MulticolumnA}[3]{&amp;\multicolumn{#1}{#2}{#3}} \newcommand{\MulticolumnB}[3]{ \multicolumn{#1}{#2}{#3}}  \NewDocumentCommand{\MulticolumnX}{m m m}{&amp;\multicolumn{#1}{#2}{#3}} \NewDocumentCommand{\MulticolumnY}{m m m}{ \multicolumn{#1}{#2}{#3}}  \begin{document} \[ \begin{tabular}{lllllll}     {a} \MulticolumnA{4}{l}{A-abcdefghijkli}\\     {a}&amp;\MulticolumnB{4}{l}{B-abcdefghijkli}\\     {a} \MulticolumnX{4}{l}{X-abcdefghijkli}\\     %{a}&amp;\MulticolumnY{4}{l}{Y-abcdefghijkli}\\     {col1}&amp;col2&amp;{col3}&amp;{col4}&amp;{col5}\\ \end{tabular} \] \end{document}   Above compiles as is, but does not compile if the one line is uncommented. How to avoid noisy renders in Cycles? How do I set a background image while I am editing? How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing? Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 Can I view a list of my "great comments", perhaps in my profile? I'm a little surprised no one asked this yet, but why isn't it possible to show your comments in descending order of "great comment" votes it has received? You can do this with upvotes on questions and answers, and since I think comments are the third major activity on SO, I'd like to view them. Even if you couldn't order them, it should at least show the number of votes on the activity#comments page of your profile... I´m having some trouble with my udev rule. At the moment, I had my udev rule setup to execute a script whenever a Android phone is connected to any USB port on the computer. But I don´t want the script to be executed from every USB port. I just want that the script to be execute in example when the phone is connected to the 4th USB port and nothing should happen when it is connected to 1st, 2nd or 3rd USB port.  How can I achieve that and how can I determine the for example 4th USB Port?  I would highly appreciate your help.  My current udev rule:  KERNELS="1-2:1.1", ACTION=="add", ENV{DEVTYPE}=="usb_device", ATTRS{idVendor}!="1d6b", ATTRS{idVendor}!="203a", ATTRS{idVendor}=="****", ATTRS{idProduct}=="****", RUN+="/usr/local/bin/tmp.sh $attr{serial}" Is time continuous or discrete? High-end compact camera with swivel screen and running on AA batteries I'm looking for a new compact camera that:   has a  is small enough to be mountable on a  and, runs on AA batteries. good flexibility   Requirement (3) stems from the fact that I may be hiking in the wild for weeks on an end, with no opportunity to change a Lithium-Ion battery. If another solution permits me to take thousands rather than hundreds photographs on full charge, I may consider that as well.  Additionally, it would be nice (but not essential) to have:   the ability to handle high dynamic range a viewfinder physical buttons — I live in a cold climate, and don't like to use touch-screens   The upper price range for compact models is still good for me, and I don't mind a pricerange of €400–€800 (but I don't mind cheaper either, if still of good quality).  I've been the happy user of several models in the , but now my PowerShot A650 IS has died. Starting from , it seems requirement (3) kills of many cameras: the , (might also fail (2)), the , the , the ... Although enough models meet requirements (1), (2), and (4), as well as the (non-essential) additional features I listed, it seems in the past 5 years, cameras have moved away from using AA batteries. What camera models still exist that meet my requirements?    Last photograph taken with my Canon PowerShot A650 IS, taking on Christmas Day 2012 in the Caldera de Taburiente, La Palma, Canary Islands, Spain.