prove that if the limit as x approaches a from the positive side of the x axis, f(x) = L, and if the limit as x approaches a from the negative side of the x axis, f(x) = L, then the limit as x approaches a of f(x) = L as well.
My teacher proposed this question as a challenge proof to do on our own and I can't seem to get it. Was Wondering if anyone could give me a hint or help me on the process to completing it.  Let $a \in \mathbb{R}$. Let $f$ be a function defined, at least, on an interval  centred at $a$, except possibly at $a$. Let $L \in \mathbb{R}$.     If $\lim\limits_{ x→a-} f(x) = \lim\limits_{ x→a+} f(x) = L$   then $\lim\limits_{ x→a} f(x) $ = $L$.    We are supposed to prove this using the "delta-epsilon" proof style. Thanks in advance!!
Add UI elements to UIScrollView? Converting ISO 8601-compliant String to java.util.Date Will my 320W PSU handle a Asus GTX 750Ti I've got a 320W PSU as far as I know and it's a Chinese PSU because I've seen the Chinese letters on it. I just had a question if I can upgrade my pc to Asus GTX 750Ti 2 GB OC.  My pc specs are:   4GB DDR3 RAM Intel Core 2 Duo E8400 3.0 GHZ 250GB HDD Onboard graphics card Re-Structuring a JSON Can I travel to Kuwait on a visa with old passport number? I have a Kuwait visa, obtained from a hotel in Kuwait, with my old passport number. But as the validity of the present passport is expiring within 6 months, I got a new passport with a new passport number. Can I travel with the same visa or do I have to obtain a new visa with my new passport number? Is there a correct resistance value for I2C pull-up resistors? How exactly to compute the ridge regression penalty parameter given the constraint? The accepted answer in  does a great job of showing that there is a one-to-one correspondence between $c$ and $\lambda$ in the two formulations of the ridge regression: $$ \underset{\beta}{min}(y-X\beta)^T(y-X\beta) + \lambda\beta^T\beta $$ and  $$ \underset{\beta}{min}(y-X\beta)^T(y-X\beta) \text{ s.t. }\beta^T\beta\leq{c} $$  The linked answer shows this in the orthogonal case.    In a general (non-orthogonal case), how can I compute $\lambda$ from $c$?  Update Here is an answer for going from $\lambda$ to $c$:   Assuming that the coefs are constrained by the penalty, $$ \beta^T\beta = c $$ and  $$ \beta = (X^TX + \lambda I)^{-1}X^Ty\\ \beta^T\beta = c = \beta^T(X^TX + \lambda I)^{-1}X^Ty $$  Still working on going the other way Calculating the expected values of the min/max of 2 random variables A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link. I can never remember how I do this because it comes up so infrequently for me. But in C or C++, what is the best way to read a character from standard input without waiting for a newline (press enter).  Also ideally it wouldn't echo the input character to the screen. I just want to capture keystrokes with out effecting the console screen. Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$? How do I install pepper-flash on Chromium? I have Chromium web browser on my Lubuntu 13.04 (pre-installed).  I use LX Terminal (pre-installed) to write commands. I am trying to download flash player on Chromium using instructions on this :  sudo add-apt-repository ppa:skunk/pepper-flash   After typing in my password, this worked. Now   sudo apt-get update   I didn't need to type in my password, as I continued on the same terminal, but got W:/ and E:/ fetch file errors  sudo apt-get install pepflashplugin-installer   I continued on the same terminal despite the fetch file errors... and they said pepflashplugin-installer could not be found.  Is the last error because of fetch file errors, or because I need to download pepflash-plugin-installer somewhere? Or is it because of something else?  I cannot download the Chrome browser, and not looking to use flash player on my Firefox web browser (installed using lubuntu software center). I have the following problem: I correctly obtain a numbered bibliography at the end but along the text the cite_key appear instead of a number. How to fix this problem?   \documentclass[11pt]{book}  \usepackage[]{biblatex} \begin{document} %some text  ... \cite{ABKM2015}... %some text  \begin{thebibliography}{99} \bibitem{ABKM2015} Arroja, F. and Bartolo, N. and Karmakar P. and Matarrese, S. (2015). \newblock The Two Faces of Mimetic Horndeski Gravity             \newblock {\em ArXiV}, 1506.08575 (2015). \end{thebibliography} \end{document}   After compiling I get ... [ABKM2015] ... while I want for example  [1]. Both options affect colors in a similar way, but I can't figure out the difference. The words in the node of tree I want to plot this graph, but I don't know how to code the words in the nodes.     Here is my code:  \documentclass{article} \usepackage{amsmath} \usepackage{titletoc} \usepackage{tikz} \usetikzlibrary{trees,positioning,fit}  \begin{document}{\allowdisplaybreaks[4]}  \tikzstyle{bag} = [text width=2em, text centered] \tikzstyle{end} = []  \begin{center} \begin{tikzpicture}[sloped] \node (a) at ( 0,0) [bag] {$\ln r_0$}; \node (b1) at ( 3,1.5) [bag] {$\ln r_{10} + \sigma_1\sqrt{\Delta t_1}$}; \node (b2) at ( 3,-1.5) [bag] {$\ln r_{10}$}; \node (c1) at ( 6,3) [bag] {$\ln r_{20} + 2\sigma_2\sqrt{\Delta t_2}$}; \node (c2) at ( 6,1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c3) at ( 6,0) [bag] {$\ln r_{20} + \sigma_2\sqrt{\Delta t_2}$}; \node (c4) at ( 6,-1.5) [bag] {$\textrm{Mean} = \ln r_{20} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2}$\\$\textrm{Mean Change} = \mu_{11}\Delta t_2 = \ln\dfrac{r_{20}}{r_{10}} + \dfrac{3}{2}\sigma_2\sqrt{\Delta t_2} - \sigma_1\sqrt{\Delta t_1}$}; \node (c5) at ( 6,-3) [bag] {$\ln r_{20}$}; \draw [-&gt;] (a) to node [below] {} (b1); \draw [-&gt;] (a) to node [above] {} (b2); \draw [-&gt;] (b1) to node [below] {} (c1); \draw [-&gt;] (b1) to node [below] {} (c2); \draw [-&gt;] (b1) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c3); \draw [-&gt;] (b2) to node [below] {} (c4); \draw [-&gt;] (b2) to node [below] {} (c5); \end{tikzpicture} \end{center}  \end{document}       Update code use  \draw [-&gt;] (a) to node [below] {} (b1.south east); \draw [-&gt;] (a) to node [above] {} (b2.north east); \draw [-&gt;] (b1.south east) to node [below] {} (c1.west); \draw [-&gt;] (b1.south east) to node [below] {} (c2.west); \draw [-&gt;] (b1.south east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c3.west); \draw [-&gt;] (b2.north east) to node [below] {} (c4.west); \draw [-&gt;] (b2.north east) to node [below] {} (c5.west); Finding $\int x^xdx$ I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:  Let $u=x^x$.  $$\begin{align} \int x^x \, dx&amp;=\int u \, du\\[6pt] &amp;=\frac{u^2}{2}\\[6pt] &amp;=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &amp;=\frac{x^{2x}}{2} \end{align}$$  But it's certain that this isn't the correct way to evaluate that, and the answer must be wrong. Graduate student in or of? In my cover letter I want to write that I am a graduate student in statistics. But I am not sure about the "in". So what is correct?     I am a graduate student in statistics.      I am a graduate student of statistics. I have a Canon 550D with 70-200 f4 and 50mm prime lenses. Although the camera has built-in 9 focal points, I sometimes face a strange problem. Here's a sample photo:    In reference to the above pic, you can see that both of the faces are NOT in focus. I used 70-200 f4 for this shot at max aperture. Still, I couldn't get both of them in focus. Why? I also try decreasing the aperture to attain a greater DoF but still, I couldn't get both of them into focus.  Another thing: Although the cam has 9 focal points, why can't I use all of them to focus, say, 9 points in a portrait shot? In manual mode, it has the option to select one of the 9 focal points. And in auto mode, it sometimes select more than one focal points.  This is just an example image. I've faced this problem zillions of times. Every time I try to create a table using MySQL Workbench 6.3 CE it adds "broken" comments at the end of the script.  Example:  I am trying to create a table called "user". This is what the table should look like:   However when I click the apply button, the script generates incorrectly:  CREATE TABLE `user` (   `userid` INT NOT NULL AUTO_INCREMENT COMMENT '',   `record_status` INT NOT NULL DEFAULT 1 COMMENT '',   `createdby` VARCHAR(75) NOT NULL COMMENT '',   `createddate` DATETIME NOT NULL COMMENT '',   `updatedby` VARCHAR(75) NULL COMMENT '',   `updateddate` DATETIME NULL COMMENT '',   `username` INT NOT NULL COMMENT '',   `password` VARCHAR(75) NULL COMMENT '',   `firstname` VARCHAR(75) NULL COMMENT '',   `lastname` VARCHAR(75) NULL COMMENT '',   `email` VARCHAR(75) NULL COMMENT '',   `facebook_userid` VARCHAR(75) NULL COMMENT '',   `email_verified` VARCHAR(75) NULL COMMENT '',   PRIMARY KEY (`userid`, `username`)  COMMENT '');   The comment in this line should not be there, so the code should be:   PRIMARY KEY (`userid`, `username`));   and the code executes when I paste it into an editor and remove this comment. However I create massive databases daily. (Each with +- 15 tables). I can't keep doing this for each table as it is time consuming.  How can I disable this feature that adds the comments to the script? I don't need them in my databases What's the best way to invite a user to answer a question on Stack Overflow? On several occasions, I was reading a question thinking "I know a guy that knows the answer to this" and that guy (or gal) also happens to be a registered user. How can I bring this to the attention of this user?