Prove that $\lim_{x \rightarrow \infty} \frac{f(x)}{x}= \alpha$ when $f$ is continuous and $\lim_{x \rightarrow \infty} (f(x+1) - f(x)) = \alpha$  The given hint suggests to consider the case of $\alpha=0$.  Given $\varepsilon \gt 0$, there exists $M$ such that $x \ge M $ implies that $|f(x+1) - f(x)| \lt \varepsilon$. Then for any $y$ which is larger than $M$, we can write it as $y=x+N$ where some $x \in [M, M+1]$.   And $f$ is continuous on $[M, M+1]$, it has a maximum $M_0$ on this interval, i.e., $|f(x)| \le M_0$ on this interval. Then $|f(x+N)| \le M_0 + N\varepsilon$. So $|\frac{f(x+N)}{x+N}| \le \frac{M_0 + N\varepsilon}{x+N}$. Then as $N \rightarrow \infty$, $\lim_{y \rightarrow \infty} \frac{f(y)}{y} \le \varepsilon$, which means that $\lim_{y \rightarrow \infty} \frac{f(y)}{y} =0$.  Is this argument valid? And I’d like to know whether there is a simple argument.
Let $f:[0,+\infty)\to\Bbb R$ be a function bounded on each finite interval. I want to show that if $\lim\limits_{x\to+\infty}[f(x+1)-f(x)]= L,$ then also $\lim\limits_{x\to+\infty}\dfrac{f(x)}x = L$
"You must wear a suit TO an interview" vs "You must wear a suit FOR an interview" In the sentence-     You must wear a suit to an interview   shouldn't the to be replaced by for? Or what's the difference between these two here? A floor is paved with rectangular marble blocks,each of length $a$ and breadth $b$.A circular block of diameter $c(c&lt;a,b)$ is thrown on the floor at random.Show that the chance that it falls entirely on one rectangular block is $\frac{(a-c)(b-c)}{ab}$  I thought over this problem,i found total number of cases as area of rectangular marble block,but i cannot find the favorable number of cases.Favorable number of cases cannot be area of rectangular marble block$-$ area of circular block,as the answer suggests.What should be the correct logic,please help. How much time have the Straw Hats spent with each other? Table and figure side-by-side with independent captions Looking to put a table and a figure side by side so that the table and figure have their own captions. A crude sketch    ------------              -----------------     TABLE    | a |  b |                     FIGURE   | c |  d |  ------------              Table 2: Caption           -----------------                            Figure 7: Caption   The closest I've gotten is:  \begin{table}% \centering \parbox{0.4\textwidth}{ \begin{footnotesize} \begin{tabular}{| l | r |} \hline some &amp; table\\ \end{tabular} \end{footnotesize} \caption{Table} \label{tab:table} } \qquad \begin{minipage}[c]{0.53\textwidth}% \centering     \includegraphics[width=1\textwidth]{awesome} \caption{Figure} \label{fig:figure} \end{minipage} \end{table}   But that gives two Table X: Caption captions. I didn't find anything in subfig.  Help? Are background jobs and foreground jobs always in running state? A job can be running or stopped (suspended).  When we are talking about foreground jobs and background jobs, are we only talking about running jobs?  Are background jobs and foreground jobs always in running state?  Does it make sense to say whether a stopped job is foreground or background? Since it's Friday and others are playing games, here's another one to take us into the weekend.  Everyone knows about those Russian nesting dolls (called matryoshka dolls, I believe). A small doll nests inside a larger, which itself is inside a larger doll, and so on.   This game does the same with words. You start with a three-letter word, add a letter to make another word, and so on. Longest word wins. The catch? Each additional letter has to form a legitimate word.  Here's one to start you off:     bur   burr   burro   burrow   burrows     Good luck. Is there a maximum possible acceleration? I'm thinking equivalence principle, possibilities of unbounded space-time curvature, quantum gravity. "The package lists or status file could not be parsed or opened" Autosegmental Representation in tikz-qtree This is what I want to achieve    and this:    But all I've been able to make with tikz-qtree is the following:  \documentclass[12pt,a4]{article} \usepackage[utf8]{inputenc} \usepackage{tipa,tikz,tikz-qtree} \begin{document}  \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ p ] ] [.R [.$\mu$ a ] [.$\mu$ : ] ] ] \end{tikzpicture} \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ p ] ] [.R [.$\mu$ a ]  ] ] \end{tikzpicture}   And this:  \begin{tikzpicture}[baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ l ] ] [.R [.$\mu$ a ] [.$\mu$ l ] ] ] \end{tikzpicture} \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ : ] ] [.R [.$\mu$ a ] [ l ] ] ] \end{tikzpicture} \end{document}   Which renders:    and    The important part here is the linking from the leaves to more than one parent (it would be nice if I had the option of making the line dashed too). Now, pst-asr doesn't quite achieve what I want it to. It's important that I have the moras ('mu's). (Also it's a pstricks package, which means I must typeset it in DVIPSPDF.) I have tried a bit of the forest-package (especially forest-GP1), but I can't make that work as I want either. Lastly, some have recommended xyling, but I find that package very difficult to use.  Any recommendations?  Thanks. Wrong Java resource bundle loaded it has the guide for 12.10 but not newer ones plz someone help Why is there a white space on the left side?  \documentclass{standalone} \usepackage{asymptote} \begin{document} \noindent \begin{asy} size(10cm,10cm); draw((0,0)--(100,100)); draw((0,100)--(100,0)); dot((50,50)); \end{asy} \end{document} Standard purpose of favorites I am wondering if my behavior regarding favorites is typical.  More importantly, if it isn't typical, does my behavior conflict with an internal metric designed to reward favorite'd questions?  I mark a question as a favorite for a few reasons:  A -- I find the question/answer(s) remarkable enough that coming back to them time and time again is of benefit to me.  B -- I am waiting to see if a sub-par quality question is going to be revised enough for me to come back and answer it.  Either way, I eventually remove my favorite.  C -- I have answered a question (of any quality) and am wanting to be notified of any competing answers.    Situations B and C are the most common for me and they typically mean I will go back and remove my favorite star later; I don't remove it because the quality of the question degraded, I merely don't want to be notified any longer.  Even stranger, I am often favoriting questions that are rather low-quality (reason B) making me wonder if low-quality posts are being rewarded via some invisible metric because I (at least temporarily) have given it what looks like a stamp of approval akin to an up-vote.    I suppose the feature being called "Favorite" makes me confused because most of the questions I select are not my favorite.  Am I accidentally messing with an invisible metric that is drawing attention to posts when that isn't desired behavior? Where are icons stored? I want to change an icon of a specific theme. Where do themes save their icons? I want to know if there is a way to switch to console mode from the boot menu.  I have an NVIDIA 1070 and every time there is an update, the GUI stops working. All I need to do is re-install the drivers; however, to do that I need to be in console mode. EDIT: I should have added more information to this question but I'll comment on all answers. In short: I've tried text instead of quiet splash I do see booting activity in text mode; however, I don't get a login screen and ctrl + Alt + F1 doesn't help because GDM keeps trying to restart. Previously I managed to kill GDM through multiple tries but is not working since the latest update. EDIT: After searching for runlevel, I've found  but I think it required me to have a running system to make changes -- please correct me if I'm wrong. What if I accidentally run command "chmod -R" on system directories (/, /etc, ...) I accidently ran  sudo chmod 755 -R /   instead of  sudo chmod 755 -R ./   I stopped it after few seconds, but now there is some problems such as  sudo: must be setuid root   How can I revert permissions back? How do I install gfortran? I am trying to install gfortran in my computer with Ubuntu 12.04 by using these commands,  sudo -i apt-get update apt-get install gfortran   but I found the program saying "unable to locate package gfortran". Then on a Ubuntu page I found some question regarding the same problem and I went through the solution from them but again I couldn't install the gfortran package.  Can anybody tell me how to install it? How to pack a sphere with cubes? I read a little bit about , but I imagine the problem changes drastically when trying to fill a sphere with unit cubes. For example, how many unit cubes could one fit into a sphere of radius five? Clearly the upper limit would be $\left \lfloor \frac{4}{3}\pi \cdot 5^3 \right \rfloor$, but I imagine there will be plenty more empty space than just a small fraction of a cube. "Not a COM32R image" error when trying to install from a USB key Not a COM32R image boot   is a message that the console infinitely keeps spitting out when I try to install Ubuntu 10.04 from a USB. I created the USB using the StartupDiskCreator in 11.04.  What is the reason for this behavior? What do I do to set it right?   I have followed the steps perfectly, as enlisted in the . Etymology of "dong" and "dongle" Dong as in ding-dong is clearly onomatopoetic as confirmed by :     ding dong        imitative of the sound of a bell, c.1560.   and similarly for :     ding (v.)       1819, "to sound as metal when struck," possibly abstracted from ding-dong, of imitative origin. The meaning "to deal heavy blows" is c.1300, probably from Old Norse dengja "to hammer," perhaps also imitative. Meaning "dent" is 1960s. Related: Dinged; dinging.   What, then, is the origin of dong to mean penis? Etymonline.com is  here:     dong (n.)        "penis," 1891, of unknown origin.   We also have a much more recent word, dongle, which has no entry at etymonline.com but  claims it is:     Apparently from dangle.   This leads me to the theory that the anatomical and onomatopoetic dongs  are not related and that the former usage, like dongle, derives from dangle. So, my questions are:   Is that so? Does dong (penis) derive from dangle? Sounds reasonable enough but I have found no definitive sources for it. Does dongle derive from dangle? It makes perfect descriptive sense for dong but the USB devices called dongles do not dangle but instead protrude quite rigidly, so I feel that dongle being derived from the cruder meaning of dong makes more sense.