Let $f: \mathbb{R} \to \mathbb{R}$ such that $\lim\limits_{x \to \infty} \frac{f(x)}{x} = c$ where $c\neq 0$ or $\infty$.  Is it true that for all such functions $f$, if $\lim\limits_{x \to \infty}(f(x+1)-f(x))$ exists, then it is equal to $c$?  Note : Say if $f(x) = x+\sin(x)$, then the condition is clearly false because the second limit does not exist. But say, we impose another condition that the second limit should exist, then is it necessary that it goes to $c$ as well?
Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$.  How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ?  I am just trying to prove the convergence, not that it converges to $l$ (that part be deduced immediately by Stolz-Cesaro and the sequential characterization of limits)
Stop making iPhone constantly change sound volume on its own? I regularly get interrupted in watching and listening to video by the volume randomly getting lowered without me asking for it. I get a system notification telling me &quot;the volume was lowered&quot;, telling me it's protecting my hearing based on the last 7 days of use. I've never been asked about enabling such a feature. I have already disabled what relevant items I can find under Sound &amp; Haptics, which is given as the source of the notification. It apparently doesn't matter though because it's not being respected. How can I make this stop? iPhone 8, iOS 14.2 \begin{eqnarray*} M&amp;=&amp; \max{\left \{a,b,\frac{a+b}{2},\right}\\ &amp;&amp; a+b,a+b/2\Bigg\} \end{eqnarray*} Calculating $\lim_{n\to\infty}\sqrt[n]{2^n-n^2}$ With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons? Solve a linear system with more variables than equations How to define a bijection between $(0,1)$ and $(0,1]$? How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective. Suppose I have BaseClass with public methods A and B, and I create DerivedClass through inheritance.  e.g.  public DerivedClass : BaseClass {}   Now I want to develop a method C in DerivedClass that uses A and B.  Is there a way I can override methods A and B to be private in DerivedClass so that only method C is exposed to someone who wants to use my DerivedClass? "OpenID 2.0 for Google Accounts is going away"; will this affect Stack Exchange? I have a very simple question. When I do this:  \catcode`\@=0 \catcode`\\=11 @mycommand{a} @catcode`@\=0 \catcode`\@=11   It works perfectly and calls \mycommand correctly. However, when I try to define \mycommand with similar trickery:  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   @textbf{Hello #1}   @catcode`@\=0   \catcode`\@=11 }   compilation stalls and gives me the * prompt. If I give it Ctrl+D, it terminates without producing any output. Indeed, it seems that changing the catcodes inside a command has no effect whatsoever, because the following works perfectly (whereas, from my understanding, it shouldn't):  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   \textbf{Hello #1} }   Can anybody please explain to me what's happening? Unable to forward search Bash history similarly as with CTRL-r Why couldn't Bond have backup at the battle of Skyfall? I think I am missing a key plot point here.   Bond had to fake-kidnap M and bring her to Skyfall in order to lure Silva there for one last stand. Why did this have to be a secret from his colleagues at MI6? Why couldn't there have been other agents there to help defend the house and take down Silva?  Also, why did he have to have Q lay a fake trail to get Silva to follow him to Skyfall when he was actually going there, and he wanted Silva to follow him? Show default image when no uploaded image Can I smoothly upgrade from one LTS to next LTS release? 14.04 LTS No Dash / Launcher after update through Software Updater Invoking JavaScript code in an iframe from the parent page In the movie Harry Potter and the Goblet of Fire, there's a scene where the students are preparing for the Yule Ball.   Professor McGonagall is instructing the boys and girls, while Filch is choosing a vinyl record to place in an enormous turntable phonograph with a horn as big as Hagrid.   This has me wondering if the wizarding world, in general, plays vinyl records. Or, if this was just a Muggle artifact that Hogwarts kept as a novelty.   Are there other instances of witches or wizards using or owning phonographs or record players in either the books or movies? If so, what are they? (Examples help to illustrate if this was a one-off or a trend) "Was" or "were" for "half a dozen" In Microsoft Word, the following sentence is flagged.  It tells me to use "was" instead of "were"     There were half a dozen books strewn about the floor.   I would think that you would use "were" since it's a quantity more than one.  You wouldn't say, "There was twelve books strewn about the floor."  Does the use of the "half" modify it somehow? I've bunch of folders in folder A. I want to move files from all those folders to A.  Or  I want to cut all files from child folders and paste them in parent folder.  How to do that? Validate date format in php Ubuntu touch - remote access to device in our company we developed an android app installed on nexus 7 tablets and on tablet we release to several customers. This app stays in foreground active 24/7 but we have problems in monitoring  the devices due to the restrictive access policies of unrooted Android devices.   We can't for example reboot the device remotely.  We are evaluating the possibility of installing Ubuntu touch to replace android on this tablets. Does Ubuntu touch offer more freedom for the remote access of the device? May I do remotely everything I could do normally with the tablet?   Moreover, are the apps developed with ubuntu sdk stable enough to run 24/7?  Thanks