If $f:(a,b)\to \mathbb{R}$ is differentiable in a point $x\in (a,b)$,   show that $\lim\limits_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$ exists and   that it is equal to $f'(x)$.   I figured that since the change in $x$ is described trough $2h$ that there is a $\delta$-neighbourhood around $x$ such that $|x-h|=|x+h|$. Sadly, I don't know how to make use of that argument in my proof. I know for a fact that the quotient describes to neighborhoods that, when $h\to 0$, squeeze the point.
If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite.  so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this point, and also $f(x)$ must be continuous at x as well  so that mean that:  $\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} =  \lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$  and that: $\lim_{x\to x_0^+}=\lim_{x\to x_0^-}=f(x_0)$  I know I should probably use arithmetic limit laws to prove this but I can't see how what I figured out could help me. any help with that?  Thanks!
Why is it called an “Indian file”? How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct? Dirac equation as Hamiltonian system Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that Dirac equation becomes Hamiltonian equation   $$\dot{\psi}~=~\{ \psi,H\}_{PB}~?$$  Of course, such Poisson bracket would be graded (super Poisson bracket), but if it exists this would explain on classical level why $\frac{1}{2}$-spinors correspond to fermions. At what point is a creature's toughness calculated? If I have  where I can sacrifice a creature and put X 1/1 token creatures onto the battlefield, where X is the sacrificed creature's toughness.  If I have a creature that is 4/4 and cast  to give it +3/+3 then sacrifice it? Do I get 4 tokens or 7?  Also what if my 4/4 creature was dealt 2 combat damage before I sacrificed it? Do I get 2 or 4 creatures?  Finally, what about counters? If my 4/4 had a +1/+1 do I get 4 or 5 tokens? Can all real polynomials be factored into quadratic and linear factors? Conjugate Variables, Noether's Theorem and QM Prove that for every odd integer n, there is a group... Prove that for every odd integer n, there is a group with exactly n elements of order 2.  I am new to abstract algebra and proofs in general. Proving existence is difficult for me, but I will show what I have.  $$n = 2k + 1$$ $$k \in \mathbb{Z^+}$$  $$\exists G \supset H = (a_1 , a_2 , ..., a_n, e)$$  $$s.t. |a_1| = |a_2| = ... = |a_n| = 2$$  My instinct is to use Lagrange's Theorem, although I would like to try another method to prove this as we have not gone over cosets in class yet. If someone could just give me a hint of how to start, it would be very beneficial. Thank you. I am currently trying to develop my modeling skills, and following a tutorial found on YouTube. The tutorial teaches how to create an anvil. At 19:10 or so into the video, a subdivision surface modifier is used to create a smooth, soft look.  However, when I do the same, mine looks very sharp and generally bad.  Any ideas why? I will include the blend file below, I appreciate your time and suggestions.    This is his when he subdivides, which is very different from mine when I do the same at the same settings. (I cannot post more pictures because I am new, but if you check the blend file I attached, you should be able to see the difference.   This is what it looks like when I subsurf without the base face. How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ?   shows the closed form of it but not the steps in solving it.  I think I need some algebraic trick..   I thought of some hyperbolic functions and their derivatives and equalities:  If I make $ t = \sinh x$, then I can reduce the integral to  $$ \int \cosh ^2 x dx $$ though I'm not sure if this is valid. But from here, I cannot go on evaluating the integral... Install apt package in ubuntu 12.04 I have mistakenly removed the default package APT from my system .  When I run command locate apt-get, I get below output:  /home/jeet/apt-get /home/jeet/Desktop/apt/apt-0.7.25.3ubuntu9.13/cmdline/apt-get.cc /home/jeet/Desktop/apt/apt-0.7.25.3ubuntu9.13/doc/apt-get.8.xml /home/jeet/Desktop/apt/apt-0.7.25.3ubuntu9.13/doc/de/apt-get.de.8 /home/jeet/Desktop/apt/apt-0.7.25.3ubuntu9.13/doc/fr/apt-get.fr.8 /home/jeet/Desktop/apt/apt-0.7.25.3ubuntu9.13/doc/ja/apt-get.ja.8 /home/jeet/Downloads/apt-get /home/jeet/Downloads/apt-get.8.gz /usr/share/bash-completion/completions/apt-get /usr/share/bash-completion/completions/slapt-get   If I run ls | grep apt-get, then I won't get any output.  Can anybody please advice how to recover apt package or install it from start? Would a Bayesian admit that there is one fixed parameter value? In Bayesian data analysis, parameters are treated as random variables. This stems from the Bayesian subjective conceptualization of probability. But do Bayesians theoretically acknowledge that there is one true fixed parameter value out in the 'real world?'  It seems like the obvious answer is 'yes', because then trying to estimate the parameter would almost be nonsensical. An academic citation for this answer would be greatly appreciated. When I enter single quote ' in terminal it goes to some other mode, and commands don't execute. What is this mode and when do we use it?  root@sai:~# ' &gt; ls &gt; ' ls : command not found root@sai:~# I found the following proof arguing for the irrationality of $\sqrt{2}$.     Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q &gt; 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 &lt; q' &lt; q$. But an easy computation shows that $(\sqrt{2} - 1)q'$ is a nonnegative integer, contradicting the minimality of $q$.   Clearly, this is a vague proof in that it expects the reader to work out the math involved on his own. I took that as an exercise and worked out the following:  Assuming $\sqrt{2}$ to be rational, let $$\sqrt{2} = \frac{p}{q}$$ where $p$ and $q$ are least possible integers.  Clearly then, $\sqrt{2} - 1$ would also be rational, $$\sqrt{2} - 1 = \frac{p-q}{q}$$ where both $(p-q)$ and $q$ are also integers.  Multiplying both sides by $q$, we get $$(\sqrt{2} - 1)q = (p-q) = q'$$ where $q'$ is an integer. Using, $2&gt;\sqrt{2} &gt; 1$ , we get $$1&gt;\sqrt{2}-1&gt;0$$ proving that $q'$ is a non-negative integer.  As stated in the proof, computing $(\sqrt{2}-1)q'$ we get, $$(\sqrt{2}-1)q' = \frac{(p-q)^2}{q}$$which is not necessarily an integer (in fact necessarily not an integer). However, the proof claims it to be.  Am I missing something or something is wrong with how I proceeded?  Here is a link to the textbook where I found this proof: , (p 32, problem 1.14).  P.S. This is my first post here and my first time with TeX. I'd also like to know if there is anything wrong with the way my question is presented. I deleted a folder in Ubuntu, how can I recover it? The folder is massive (390GB) and contains many different files. The file system is ext4  I was about to copy this file to another hard drive using gksudo nautilus so I didnt have do a massive chmod operation on the destination hard drive, but then I accidentally hit move to trash, which in gksudo nautilus means permanent deletion. I checked the trash but it's not even an operation...  I have seen tools like testdisk and other recovery tools but none that can recover and whole folder because its made up of different files.   I don't think that this would work but if I deleted the partition could I just recover the whole partition at once?  Or would this be wrong because i have already deleted the partition's content? This is my first job in Linux IT.  I don't have a backup... I was trying to make one because the old backup drive died :( How do we delete subscribers (emails) from Auto suppression list using SSJS. Also, Auto suppression list maintain at SF end or application level. Hide details of a chapter from TOC and list of tables, figures and algorithms Lat/Long co-ordinates are not plotting onto EPSG:3857 OSM basemap correctly Excuse me if this is a stupid question, but I've honestly tried to find the answer in a few threads here on SE, and I cannot seem to follow the suggested steps correctly.    Project is in  (WGS 84 / Pseudo-Mercator) BaseMap (OpenStreetMap) layer is set to EPSG:3857 Vector data has been imported from CSV, and has Lat/Long coordinates, set to EPSG:3857.    The layer appears right in the middle of the map, around 0,0 obviously.   Why is this, what am I doing wrong? What package should I use to typeset units? I know there are a few packages related to typesetting units:   unitsdef units, which can be seen as a "base package" used by unitsdef SIunits siunitx, which is the successor to SIunits maybe others? like cjwunits and hepunits   What is the difference between them? Do they also work inside Math mode? And, more importantly, which one should I use?  Suggestion for people trying to answer this question: show some examples of how to typeset both simple and complex units. For instance, try typesetting these items:   100 Celsius 3 x 10^5 km/s gravitational constant is measured in N m^2/kg^2 10 kHz = 1 s^{-4} sin(x) meters   I think these examples cover most needs when writing scientific texts: simple measurements (value + simple unit), values written in scientific notation, simple fractional units, complex fractional and squared units, units without any value accompanying them, units with arbitrary exponents, and arbitrary math code instead of a simple value. What exactly are the spikes, or peaks and valleys, caused by in pictures such as these    Wikipedia states that "From the point of view of magnetic energy, peaks and valleys are energetically favorable", but I'm not exactly sure what that means, or how it works. Why isn't the fluid just happy being attracted to the magnet in a semi-sphere shape? Could it be that it gains a magnetic charge, so it attempts to repel itself at the same time? Just pulling at straws here. Can I choose to delay my turn in combat? In D&amp;D 4e there was an option to Delay your initiative:     Perform your actions as desired and adjust your initiative to your new   position in the order.   Does this rule to change initiative order still exist in 5th edition? I cannot find it in the PHB.