The following statement appeared in my textbook:     Consider $h_{n}(x)=x^{1+\frac{1}{2n-1}}$ on the set $[-1, 1]$, for a    fixed $x \in [-1, 1]$, we have $\lim_{n \rightarrow \infty} h_n(x) =  |x|$.   Can someone explain why $\lim_{n \rightarrow \infty} h_{n}(x) = |x|$? Specifically, where did the absolute value sign come from? What limit laws/rules are used to obtain this result?  I would have thought $\lim_{n \rightarrow \infty} h_n(x) = x \lim_{n \rightarrow \infty} x^{\frac{1}{2n-1}} = x$?  EDIT: I had a look at the other question suggested, but my question is different because I want to know the exact limit laws used in each step of the justification that $\lim_{n \rightarrow \infty} h_n(x) =  |x|$.
Okay, maybe I'm missing something obvious but consider $h_n(x) = x^{ 1+ \frac{1}{2n-1}}$ on the set $[-1,1]$. Now consider a fixed $x \in [-1,1]$, we have:  $$\lim_{n\rightarrow \infty} h_n(x) = x \lim_{n \rightarrow \infty} x^\frac{1}{2n-1} = x$$  But my book says it's $|x|$? Why?
How do I kill background processes / jobs when my shell script exits? I am looking for a way to clean up the mess when my top-level script exits.  Especially if I want to use set -e, I wish the background process would die when the script exits. How to solve a difference equation? So I have the difference equation $4y_{k+1} = 2y_k$.  I need to solve this equation, first with the initial condition $y_0 = 0$ and then with the initial condition $y_0 = 2$.  The professor wrote on the answer sheet that applying iteratively the recurrence relation it is easy to check that:  $y_k= (1/2)^k y_0 $.  Since I missed the lecture where he explained this, can somebody explain how to come about with this question?  I imagine that once I get the equation, then the initial condition $y_0 = 0$ gives us the particular solution $y_k = 0$. Thus the other initial condition $y_0 = 2$ is verified by the solution $y_k = 2(1/2)^k$.  Any help on how to get to the equation $y_k= (1/2)^k y_0$ will be appreciated. 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Set an alias on Bourne shell I am using a scrtipt with instruction 'SH -c command', where command is an alias from bash, this alias can not be interpreted into Bourne shell. ¿Do you have any idea for set this alias into Bourne shell and getting that it works in the new shell? Thanks everyone ! How to boot WUBI installed Ubuntu within Windows 8? I have Windows 8 Release Customer Preview on my laptop. I tried to install Ubuntu on it with Ubuntu Windows installer (I have always used this to install, although that was on Windows 7), it installs and asks to reboot, then immediately Windows 8 boots up.  I do not have the option to choose what OS to boot. How can I boot to Ubuntu? VM is not exactly what I would like to use, but it is an option, however a last resort. Any help? I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:  Proof. Let $\mathfrak s$ denote the scalar matrices. 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Beamer: Change only several bullets in the list I would like to change the bullets in the second itemize to arrows.  \begin{alertblock}{The Aim of this Research}      \begin{itemize}         \item Item 1         \item Item 2             \begin{itemize}                 \item Item 2-1                 \item Item 2-2             \end{itemize}     \end{itemize}  \end{alertblock}   Form above, I would like to make:  ・Item 1  ・Item 2    =&gt; Item 2-1  (change bullet to arrow)    =&gt; Item 2-2   I've tried inserting \renewcommand{\labelitemii}{$\rightarrow$} before and after the second \begin{itemize} and , but they don't work. How can I identify hardware components in a laptop? I just reinstalled my Windows and now I don't have all the hardware components recognized. I have an  laptop and  website seems to have all the necessary drivers for my model. The problem is that I don't know exactly which components I have. For example, in my laptop video card case the site has available drivers for ATI and Intel chips.   How do I know what are the right drivers for my machine? Is there a way to determine the respective canonical status of Tolkien's Middle-earth related works? It is almost always difficult to determine the canonicity (if that's a word) of different works within a fictional universe, but the problems are even more complicated in regards to Tolkien's Middle-earth/Arda related material.  It is safe to say that LotR, the second edition of The Hobbit, and probably the published Silmarillion are canon.  But even the published Silmarillion is, in large part, the work of Christopher Tolkien, although to a lesser degree than it is the work of his more famous father.    When we look at the rest of the corpus, things become even more opaque.  Unfinished Tales contains different versions of the same stories.  The same is true, in even greater degree, of The History of Middle-earth.  These volumes might be considered an almost equal endeavor on the parts of Christopher and J.R.R Tolkien.    Is there a tentative canon of Tolkien's Middle-earth related works, and if not, is it possible to create one?    With such a large body of work, of varying degrees of canonicity, it would be very helpful to have something like the now-outdated levels of canon associated with the Star Wars universe, ranked by a series of letters.  Has anyone put together such a system for Tolkien's Middle-earth related works? How to get over "device or resource busy"? I tried to rm -rf a folder, and got "device or resource busy".  In Windows, I would have used LockHunter to resolve this. What's the linux equivalent? (Please give as answer a simple "unlock this" method, and not complete articles like . Although they're useful, I'm currently interested in just ASimpleMethodThatWorks™) Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$. I have a scenario. (Windows Forms, C#, .NET)   There is a main form which hosts some user control. The user control does some heavy data operation, such that if I directly call the UserControl_Load method the UI become nonresponsive for the duration for load method execution. To overcome this I load data on different thread (trying to change existing code as little as I can)   I used a background worker thread which will be loading the data and when done will notify the application that it has done its work.   Now came a real problem. All the UI (main form and its child usercontrols) was created on the primary main thread. In the LOAD method of the usercontrol I'm fetching data based on the values of some control (like textbox) on userControl.    The pseudocode would look like this:  CODE 1  UserContrl1_LoadDataMethod() {     if (textbox1.text == "MyName") // This gives exception     {         //Load data corresponding to "MyName".         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage.     } }   The Exception it gave was     Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on.   To know more about this I did some googling and a suggestion came up like using the following code  CODE 2  UserContrl1_LoadDataMethod() {     if (InvokeRequired) // Line #1     {         this.Invoke(new MethodInvoker(UserContrl1_LoadDataMethod));         return;     }      if (textbox1.text == "MyName") // Now it wont give an exception     {     //Load data correspondin to "MyName"         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage     } }   BUT BUT BUT... it seems I'm back to square one. The Application again  become nonresponsive. It seems to be due to the execution of line #1 if condition. The loading task is again done by the parent thread and not the third that I spawned.  I don't know whether I perceived this right or wrong. I'm new to threading.  How do I resolve this and also what is the effect of execution of Line#1 if block?  The situation is this: I want to load data into a global variable based on the value of a control. I don't want to change the value of a control from the child thread. I'm not going to do it ever from a child thread.   So only accessing the value so that the corresponding data can be fetched from the database. Easy curves in TikZ I find myself needing to draw lots of elegantly curved paths in TikZ. Ideally, I'd just specify a series of points, and TikZ would calculate the extra data itself to draw a nice series of curves passing smoothly through these points, perhaps with an optional "looseness" parameter that I could specify. But the only way I can find to draw nice curves is by explicitly giving control points, or by manually specifying in and out angles.  I can think up a simple algorithm to do this, which would certainly be within TikZ's power to perform: just choose the in and out angles in a simple fashion based on the relative angles between each adjacent pair of line segments.  Is something like this already built-in? Or can someone cook something up that does the job?  Edit: Jake has provided an answer using the plot [smooth] functionality. This is almost perfect! But it can't do what I need, because it doesn't let me specify tangent angles manually where needed, which is especially important at the beginning and end of the curve. 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Reload activity in Android