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Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Problem:   If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|&gt;1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$.   This approach is definitely wrong:    \begin{align} b&amp;=\lim_{x\to 0} \frac{f(ax)-f(x)}{x}\\ &amp;=\lim_{x\to 0} \frac{f(ax)-f(0)-(f(x)-f(0))}{x}\\ &amp;=af'(0)-f'(0)\\ &amp;=(a-1)f'(0) \end{align}  I will show you a case why this approach is wrong:     \[f(x)= \begin{cases} 1,&amp;x\neq0\\ 0,&amp;x=0 \end{cases}\]   $\lim_{x\to0}\dfrac{f(3x)-f(x)}{x}=\lim_{x\to0} \dfrac{1-1}{x}=0$   but $\lim_{x\to0}\dfrac{f(3x)}{x}=\infty$,$\lim_{x\to0}\dfrac{f(x)}{x}=\infty$   Does anyone know how to prove it? Thanks in advance! How did "but" mean "only"? : Old English butan, buton &quot;unless, except; without, outside,&quot; [...]  I don't know Old English. From the étymons overhead, how did but  to mean    |adverb| = 1. No more than; only  ? What can explain the semantic shift? I didn't understand , because English is not my first language  :   In sense: Only. An elliptic development of the conjunction: see .  a. By the omission of the negative accompanying the preceding verb (see ), but passes into the adverbial sense of: Nought but, no more than, only, merely.    I understand Shakespeare's use in . Thank you all. There is a nice property of normal spaces, namely, closed disjoint subsets can be separated by continuous functions into $\mathbb R$.   Then you ask yourself, what about Hausdorff spaces?, are all Hausdorff spaces completely Hausdorff?   Well, if we choose our separating space $X$, that is, the space such that for any Hausdorff $Y$ and any distinct $a,b\in Y$ there is some continuous $f:Y\rightarrow X$ with $f(a)\neq f(b)$, to be completely regular and Hausdorff, this cannot happen by an example of E. Hewitt, Hewitt’s Condensed Corkscrew, call it $\mathbb H$; as any continuous function from $\mathbb H$ into $\mathbb R$ is constant, and any Hausdorff completely regular space $X$ can be embedded into a product of closed intervals $\prod_{i\in I}[0,1]$, and thus any continuous $f:\mathbb H\rightarrow X\subseteq \prod_{i\in I}[0,1]$ is constant, as each $\pi_i\circ f$ is constant, so our candidate to be the separating space $X$ cannot even be $[0,1]$.  My question is, does there exist a Hausdorff space $X$ such that for all Hausdorff spaces $Y$, and any distinct $a,b\in Y$ there exists a continuous $f:Y\to X$ such that $f(a)\neq f(b)$?  Thanks. What does Geordi see on the holodeck? We've seen how he interprets people and things with his VISOR, but if the holodeck is all force-fields and holographic imaging, how does he interact in a similar way to other members?   What does the holodeck look like to Geordi LaForge using his VISOR? I'm trying to install queXS cati app on my Ubuntu desktop and I installed MySQL server and PHP 5 and I cannot login into MySQL server as root without password:  mysql -u root    it says  ERROR 1045(28000) : Access denied for user 'root@localhost' (using password: no )   But it's okay when I enter mysql -u root -p   I can't figure out what the problem is. argument about fallacy of diff(M) being a gauge group for general relativity I want to outline a solid argument (or bulletpoints) to show how weak is the idea of diff(M) being the gauge group of general relativity.  basically i have these points that in my view are very solid but i want to understand if there are misconceptions on my part that i'm simply not getting and if its so, i ask for help to make the case more solid, or understanding why it doesn't apply (to gravity):    (thanks to Raymond Streater for making that point completely clear) gauge invariance in electrodynamics is an observation that physical observables are unchanged after a gauge transformation without changing coordinate frame ( we are ask to believe that in gravity someone did the same? that is, someone made the observation that physical observables are unchanged after a diffeomorphism-gauge-transformation, only to later argue that because of this, that there are no physical observables to begin with, that doesn't make a lot of sense, to not say that its just plain stupid circular argument ) . The invariance is of course broken when the theory is quantized and $\hbar$ makes an appearance, because it assumes a preferred scale for certain energies. The key point here is: classical gravity is not special regarding having diff(M) as a symmetry group from bulletpoints 2 and 3, if i cannot infer that Diff(M) is a gauge-invariance of electrodynamics, the same should apply to gravity   For this question, i would say that a valid answer would either disprove any of the arguments as fallacies themselves (hence showing a solid argument why gravity is special and diff(M) is without a doubt its gauge group), or improve the argument for making it bullet-proof (sorry for the pun) Can I use mirroring to make a copy of my Database? Remove linux version 3.19.31 How to fix Footnote position at the bottom of the page \footnote[1]{\nameref*{8.1.16} is an अधिकार सूत्र it goes upto \nameref*{8.3.55}} \&amp; अङ्गाधिकार (अङ्गस्य)\footnote[2]{\nameref*{6.4.1} is an अधिकार सूत्र it goes upto \nameref*{7.4.97}}  I am using footnote as shown above. However the footnote appears to be floating and when I don't have much in the page it appears in the middle of the page. I got by so far with \vspace{} but I think there must be a way to position this at the bottom of the page despite whats in the page. Thanks for your help Why on the first page Google says there are thousands of results but on the last page there are less than a hundred? How do people who do not speak Indian English "prepone" things? Unable to lock the administration directory (/var/lib/dpkg/), are you root? I get this error in Lubuntu 12.04 when I try to update in terminal:  $ apt-get update E: Could not open lock file /var/lib/apt/lists/lock - open (13: Permission denied) E: Unable to lock directory /var/lib/apt/lists/ E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root? Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$ Knowing that $$ \frac{a_0}{1} + \frac{a_1}{2} + \frac{a_2}{3} +\cdots + \frac{a_n}{n+1} =0$$ Prove that  $$ a_0 + a_1x + a_2x^2 + \cdots + a_nx^n = 0$$ has at least one real solution.  I suspect that it's proven with the intermediate value theorem. But can't find two numbers that satisfy it. What is the significance of allowing nilpotents in scheme theory? If $a$ and $b$ are two roots of $x^4 + x^3 - 1 = 0$ prove that $ab$ is a root of $x^6 + x^4 + x^3 - x^2 - 1$. Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$? I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the smoothing-out happen so relatively quickly?  $Z_2$:     $Z_3$:    (images shamelessly stolen from John D. Cook's blog: )  Why doesn't it take, say, four uniforms? Or five? Or...? Analyzing limits problem Calculus (tell me where I'm wrong). I came accross: $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}}$$  I tried the following please tell me where I 'm wrong:  $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}} $$  $$\text{(Dividing by }x)$$   $$=\displaystyle\lim_{x \to\ 0}\dfrac{ \cos x - \dfrac{\log (1 + x)}{x}}{x} $$  $$=\lim_{x \to\ 0}\frac{\cos x - 1}{x} $$  $$=\lim_{x \to\ 0}\frac{-2\sin^{2} \dfrac{x}{2}}{x} $$  $$=\lim_{x \to\ 0}\frac{-2x\sin^{2}\dfrac{x}{2}}{(\dfrac{x}{2})^{2}\times 4} $$ $$\lim_{x \to\ 0}\dfrac{-x}{2}=0$$ But Answer given is $\dfrac{1}{2}$  Please Help. How do I close a connection early? I'm attempting to do an AJAX call (via JQuery) that will initiate a fairly long process. I'd like the script to simply send a response indicating that the process has started, but JQuery won't return the response until the PHP script is done running.  I've tried this with a "close" header (below), and also with output buffering; neither seems to work. Any guesses? or is this something I need to do in JQuery?  &lt;?php  echo( "We'll email you as soon as this is done." );  header( "Connection: Close" );  // do some stuff that will take a while  mail( 'dude@thatplace.com', "okay I'm done", 'Yup, all done.' );  ?&gt; What's the point of Hamiltonian mechanics? I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I specially appreciated the freedom you have when choosing coordinates, and the fact that you can basically ignore constraint forces. Of course, most simple situations you can solve using good old $F=ma$, but for more complicated stuff the whole formalism comes in pretty handy.  Then in the second half we switched to Hamiltonian mechanics, and that's where I began to lose sight of why we were doing things the way we were. I don't have any problem understanding the Hamiltonian, or Hamilton's equations, or the Hamilton-Jacobi equation, or what have you. My issue is that I don't understand why would someone bother developing all this to do the same things you did before but in a different way. In fact, in most cases you need to start with a Lagrangian and get the momenta from $p = \frac{\partial L}{\partial \dot{q}}$, and the Hamiltonian from $H = \sum \dot{q_i}p_i - L$. But if you already have the Lagrangian, why not just solve the Euler-Lagrange equations?  I guess maybe there are interesting uses of the Hamiltion formalism and we just didn't do a whole lot of examples (it was the harmonic oscillator the whole way, pretty much). I've also heard that it allows a somewhat smooth transition into quantum mechanics. We did work out a way to get Schrödinger's equation doing stuff with the action. But still something's not clicking.  My questions are the following: Why do people use the Hamiltonian formalism? Is it better for theoretical work? Are there problems that are more easily solved using Hamilton's mechanics instead of Lagrange's? What are some examples of that? Goal: I want to be able to securely use the internet via my home PC while my notebook is connected to an open hotspot/access point.  I do know that I can use a SSH tunnel/SOCKS proxy, but I don't want to fiddle around with applications (making them use it, if even possible). I guess what I need is an OpenVPN setup, so I'm looking for a detailed guide on how to:   Install and setup the OpenVPN server Setup the OpenVPN client (NetworkManager)   Ubuntu versions this should work on are 10.10 and 11.04.