Let $f:[0,\infty )\rightarrow \mathbb{R}$ be an increasing function, such that $\lim_{x\rightarrow \infty }\frac{1}{x^{2}}\cdot \int_{0}^{x}f(t)dt=1$.  Prove that exists $\lim_{x\rightarrow \infty }\frac{f(x)}{x}$ and calculate this limit.    If $f$ would be continuous, we'd have $1=\lim_{x\rightarrow \infty }\frac{1}{x^{2}}\cdot \int_{0}^{x}f(t)dt=\lim_{x\rightarrow \infty }\frac{f(x)}{2x}$, therefore $\lim_{x\rightarrow \infty }\frac{f(x)}{x}=2.$ But we don't know if $f$ is continuous or not, and I wasn't able to find any other idea.
Let $f$ be a real, continuous function defined on $[0,\infty)$ such that $xf(x)$ is increasing for all sufficiently large values of $x$. Show that if   $$\int_0^x f(y)\,dy \sim Ax^\alpha \quad \left(\,x\to \infty\right)$$  for some positive constants $A$ and $\alpha$, then  $$f(x)\sim \alpha Ax^{\alpha -1} \quad \left(\,x\to \infty\right).$$  Clearly, I have to use differentiation somewhere, but I don't know how to manipulate $\lim_{x\to \infty}\frac{\int_0^x f(y)\,dy}{Ax^\alpha}=1$ to get the desired result.  From suggestions given, I know that by L'hospital's rule, it's enough to show that the limit $\lim \frac{f(x)}{\alpha Ax^{\alpha -1}}$ exists, and this limit equals $ \lim \frac{xf(x)}{\alpha Ax^{\alpha}}$. From here, I'll need to use the given assumption that $xf(x)$ is eventually increasing. But how can I show the existence of the limit based on these facts?  I would greatly appreciate any solutions, hints or suggestions.
Don't remember the exact start to the story, but it went something like "walking along a path" or something, and when he hears what sounds like a shotgun blast behind him, he then feels something strike him in the butt. He passes out, and when he awakens, he discovers that the "shotgun blast" was actually a miniature spaceship that exploded and shot miniature aliens into his body. The aliens prove to be very beneficial because they clean up all the toxins and other poisons in his body. They can be transmitted through kissing or sexual activity.  Any help with the name of the story would be greatly appreciated.  Dan Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$ In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL. "Requires 50 reputation" to comment on deleted posts So I recently saw add comment (requires 50 reputation) on a deleted answer. After looking around a bit, it seems to be on all deleted posts now (only deleted posts, normal posts still say just add comment).  I'm sure it wasn't there before (I think it just said add comment (unclickable) or possible nothing).  It is also on , presumably the root cause is the same.  And yes, I do have more than 50 reputation (in case anyone was wondering).  Just to clarify, requires 50 reputation obviously shouldn't be there for users with 50 or more reputation. Choices:   Display nothing Unclickable add comment - probably not ideal Unclickable add comment (disabled for deleted posts) or something along those lines   FYI - You may be somewhat confused right about now if you can't  yet. What is the right way to consistently mangle bash terminal? I'm on OSX 10.11.1 and occasionally my bash terminal gets mangled. It often happens when I accidentally cat a binary file. The result can be seen on the image below. The output becomes weird, and I can't type ascii characters anymore.    Even though this happens occasionally, I couldn't find a way to consistently reproduce the issue. Online search recommends doing cat /bin/*, but that works sporadically, only after a couple dozen tries.  I want to do this so I can find an easy solution how to handle this in tmux.  How do I consistently get bash to a "mangled" state? Is there maybe a magic unicode character that can do this? Can a polymorphed creature use class features that don't require speech or hand gestures? The Polymorph spell states (in part):     The target’s game statistics, including mental ability scores, are   replaced by the statistics of the chosen beast. It retains its   alignment and personality.      The creature is limited in actions it can perform by the nature of its   new form, and it can’t speak, cast spells, or take any other action   that requires hands, or speech.   So, can a polymorphed Paladin use their Divine Smite class feature?  It expends a spell slot, but it is not a spell, and as far as we can tell does not require hands or speech.  We know (based on the 10th level Transmutation Wizard feature) that a wizard is supposed to be able to polymorph herself, and we can assume that she is also supposed to be able to maintain concentration needed to stay in the new form (otherwise it would be a pretty useless ability).  This is (as far as I can tell) not something any of the beasts eligible as polymorph targets can normally do.  So it is clear that the polymorphed creature maintains some abilities from her old form.  The question is which ones.    The Druid's Wild Shape ability explicitly says you can use class features, while Polymorph does not, suggesting that you can't use class features while polymorphed.  However, Wild Shape also explicitly says that you can continue to concentrate on a spell (see below, emphasis mine), and Polymorph does not, yet we believe that a polymorphed creature can continue to concentrate on the polymorph spell.  So the fact that Wild Shape explicitly allows something and Polymorph is not explicit about it does not necessarily mean that the polymorphed creature cannot do those things.  From Wild Shape rules:     You can't cast spells, and your ability to speak or take any action   that requires hands is limited to the capabilities of your beast form.   Transforming doesn't break your concentration on a spell you've   already cast, however, or prevent you from taking actions that are   part of a spell, such as call lightning, that you've already cast.   So, since Polymorph is silent on the question of whether or not class abilities can be used, and states only that the polymorphed creature cannot perform actions that require hands or speech, how can we best determine whether (RAI) polymorphed creatures maintain any of their old abilities beyond spell concentration? Is it possible to export a single object from an entire scene? Equality in an integral estimate Let $u,v\in W^{1,p}(\Omega)$ with $1&lt;p&lt;\infty$ and $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$. Suppose, we have $$\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\,dx=\left(\int_{\Omega}|\nabla u|^p\,dx\right)^\frac{p-1}{p}\left(\int_{\Omega}|\nabla v|^p\,dx\right)^\frac{1}{p} $$ Suppose the equality $$ \int_{\Omega}|\nabla u|^p\,dx=\int_{\Omega}|\nabla v|^p\,dx $$ holds. Then can we say that $u=v$ in $\Omega$? I tried in the following way: Since the first given equality, which is the equality in Holder's inequality holds, and by the second given equality holds, we have $|\nabla u|=|\nabla v|$ in $\Omega$. Then I am unable to proceed. Can someone please help to give an argument? I'm using uninstall.php for my plugin to delete all the traces my plugin did to a site on uninstall (delete). I did it in a good way, but even after the deletion there are some fields in the options table:   uninstall_plugins recently_activated _site_transient_update_plugins   in these options there are multiple values from other plugins too.  How can I take care of such traces, how can I remove all those traces too?  I checked WooCommerce and found they leave no traces after there deletion, if the "delete data" checkbox is checked.  How can I do so? How do you mark a struct template as friend? How can I have 2 different mod folders with the new Minecraft launcher? Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ If light rays obey to the wave equation, why can they be thought as straight lines? I'm a newbie with physics but I'm wondering how a ray of light can essentially be represented. I have always known that a ray of light proceeds in a straight line until it encounters another object (or material) that refracts/reflects it. But a light ray should be part of an electro-magnetic wave, is this correct? If it is so, it should obey to the  and this doesn't seem to me to describe a straight line ray.  I'm having problems visualizing how light is emitted and how it relates with the wave equation. Can someone with a clear understanding of the problem explain it to me in simple terms? You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i. $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right), $$ where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ (e.g. see here ) Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}.   $$ The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0. $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this? Quantum mechanics - how can the energy be complex? In section 134 of Vol. 3 (Quantum Mechanics), Landau and Lifshitz make the energy complex in order to describe a particle that can decay:  $$ E = E_0 - \frac{1}{2}i \Gamma. $$  The propagator $U(t) = \exp(-i H t)$ then makes the wavefunction die exponentially with time.  But also, $H$ is non-Hermitian.  My question: Do we have to modify the basic postulates of quantum mechanics (as described by Shankar, say, or the earlier sections of Landau &amp; Lifshitz) to describe unstable particles? CASE WHEN with GROUP BY I have a table that looks like this (oversimplified case of the actual case to make my point)  CREATE TABLE sales (    id int,    currency VARCHAR(5),    price int    shop_id int );   Now I need to make a query that will give me something like this table, grouping by shops yet giving sum of sales per currency My problem is when I make currency column non aggregate it fails because my query becomes non standard. When I add currency to group by then each currency creates new row, something I don't want. I have run out of option and would appreciate any help or pointer to the right direction.  I use MySQL but would appreciate if I would get the solution that works with more databases (PostGreSQL being at the top of the list after MySQL)  Query that works with undesired results  SELECT   shop_id,           (CASE WHEN currency= "GBP" THEN SUM(price) ELSE 0 END) AS POUND,           (CASE WHEN currency= "USD" THEN SUM(price) ELSE 0 END) AS USDOLLAR GROUP BY shop_id, currency   The data that should be produced by the query  shop_id     |   POUND   |   USDOLLAR     -----------------------------------------     1       |   400     |   300 -----------------------------------------     2       |   250     |   100 -----------------------------------------     3       |   400     |   100 ----------------------------------------- How do I redirect in a class sub-classing the EventSubscriber class? How to generate the random default "gravatars" like on Stack Overflow? How does Stack Overflow auto-generate the gravatars for those users who do not upload a picture or who have not created a Gravatar icon?    Where do these random avatars come from, and how can I generate them for my own site? What is the best regular expression to check if a string is a valid URL? How can I check if a given string is a valid URL address?  My knowledge of regular expressions is basic and doesn't allow me to choose from the hundreds of regular expressions I've already seen on the web. For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$