$$\lim_{x \to 0}x^x$$   I know the answer is one but I have no idea how to get there.  I tried taking a natural log and I think I need lhopitals rule but I keep going In circles.
I know when evaluated it gives one but I can't figure out how to prove it.  Can anyone help? I believe it requires L'Hopitals rule and taking a natural log but i cannot figure out the exact math.
Which are the safety recommendations for soldering? Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common.   $$\sum _i ( \mathbf F _i - \dot{\mathbf p}_i) \cdot \delta \mathbf r _i = 0$$  $$S[q(t)] = \int ^{t_2} _{t_1} \mathcal L (q,\dot{q},t)dt$$  After playing with d'Alembert's principle  we find that we can rewrite the whole thing as $$\sum _i \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i}-Q_i\right] \delta q_i$$  This can further be rewritten under certain conditions so we get the exact form of the E-L equation.   It seems to me that both ways of arriving at the result are fundamentally different.   A function must obey the E-L equations in order to minimize the action over a path, but when we look at the virtual work, it appears that they come from the fact that (quoting Goldstein) "particles in the system will be in equilibrium under a force equal to the actual force plus a 'reversed effective force'."  I think I understand the principle of stationary action, I can see how it leads to the E-L equations, but d'Alembert's Principle seems so arbitrary, I can't see any motivation for it. Backup whole OS How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago? I was reading through the Wikipedia article on  and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the  exactly says      The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.   Now I come to understand that the  singularity is believed to be around 13.8 Billion years ago.  So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?   I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?  Edit: My Bad! Here's how..  A simple google search led me to  which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Is this defined by the language? Is there a defined maximum? Is it different in different browsers? Filling HTML  dropdown list in JSP with values fetched from database in Servlet I'm trying to make a command block that spawns an item in front of you if you are holding an item with a specific name. I've got these two working scripts;  /testfor @p[r=10] {Inventory:[{tag:{display:{Name:"Item Name"}}}]}  /testfor @p[r=10] {SelectedItemSlot:0,Inventory:[{Slot:0b,­id:"minecraft:diamond_sword"}]­}   but I can't get them together. The block should only look for an item in the active slot. If $a,b,c$ are positive real numbers, not all equal, prove that  $$\dfrac{bc}{b+c}+\dfrac{ca}{c+a}+\dfrac{ab}{a+b}&lt;\dfrac{a+b+c}{2}$$  Progress: $\frac{bc}{b+c}-\frac{b+c}{2}=$ Are there any holidays in Middle Earth, except Yule, Lithe, New Year in Gondor and Durin's Day? apt-get update fails to fetch files, “Temporary failure resolving …” error Why is "cupboard" pronounced with a silent "p"? According to Google at least, the word "cupboard" originated in late Middle English as denoting a board that held cups. Since then, the word has evolved to mean a kind of cabinet.  My question is, given its origin and spelling, why do we pronounce "cupboard" with a silent "p"? Has the pronunciation simply evolved because "cup-board" is too awkward to say, or is there a deeper pronunciation rule that I'm not aware of? C++ equivalent of java's instanceof What is the preferred method to achieve the C++ equivalent of java's instanceof? Slow math typing I have been using LaTeX for quite some time, but I find the math writing to be a bit frustrating. It seems to take me a relatively long time to code-write long math expressions, and sometimes it's hard to see where { start and end.  I am interested in writing as fast as possible.  Is code-typing each symbol really the fastest way? Is there any good program (in the spirit of Word's MathType) that enables producing math expressions by button clicking, that you recommend? In an extract I am dealing with, I have 2 datetime columns. One column stores the dates and another the times as shown.  How can I query the table to combine these two fields into 1 column of type datetime?  Dates  2009-03-12 00:00:00.000 2009-03-26 00:00:00.000 2009-03-26 00:00:00.000   Times  1899-12-30 12:30:00.000 1899-12-30 10:00:00.000 1899-12-30 10:00:00.000 Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited Third grade spiral in TikZ - dimension too large I'm trying to draw a smooth third grade spiral (r^3 = a^3\cdot \varphi) over $60\times 60$mm grid and I'm getting the "Dimension too large"-error if I set $a &gt; 320$ but I need to fill the grid area completely:  \begin{tikzpicture}[x=1mm, y=1mm]      \draw[step=1, line width=0.1mm, opacity=0.3] (-30, -30) grid (30, 30);      \draw [domain=0:50, variable=\t, samples=1000]          plot ({\t r}:{pow(320*\t, 1/3)}); \end{tikzpicture}     Is there any way to avoid this error? How do you check if a JavaScript Object is a DOM Object? I'm trying to get:  document.createElement('div')  //=&gt; true {tagName: 'foobar something'}  //=&gt; false   In my own scripts, I used to just use this since I never needed tagName as a property:  if (!object.tagName) throw ...;   So for the second object, I came up with the following as a quick solution -- which mostly works. ;)  The problem is, it depends on browsers enforcing read-only properties, which not all do.  function isDOM(obj) {   var tag = obj.tagName;   try {     obj.tagName = '';  // Read-only for DOM, should throw exception     obj.tagName = tag; // Restore for normal objects     return false;   } catch (e) {     return true;   } }   Is there a good substitute? Why are Perl 5's function prototypes bad? Should I use the first or last bits from a SHA-256 hash? Adding an error on a dynamic referenced field Is there a way to add an error on a dynamically referenced field? Let's take this simple example:  opp.get('Name').addError('Invalid name');   This is not working, I'm looking for a similar minded way.  Thanks