Prove:      If $f$ is a real-valued differentiable function such that $\lim_{x\to \infty}f'(x) = 0$ then $\lim_{x\to \infty}(f(x+1)-f(x))=0$.   Proof: We know $\lim_{x\to \infty}f'(x) = 0$.   (1)  By the mean value theorem there exists $c\in [x, x+1]$ such that $f'(c) = \dfrac{f(x+1)-f(x)}{1}$  $\iff \lim_{x\to \infty} f'(c) = \lim_{x\to \infty} f(x+1)-f(x) = 0$ by (1).  End of Proof.
Having some trouble proving this:     If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show: $\lim _{x\to \infty } (f(x+1)-f(x))=0$   Attempt: from $\int \lim _{x\rightarrow \infty } f^\prime(x) \mathrm{d}x=0$ we can say that $f^\prime(x)=const$ would we then have to show that  $\lim _{x\to \infty } (f(x+1))$ is also a constant? From there if we show that both are equal then the difference is zero. I am on the right track?  Thank you for the help.
How do you correctly say large numbers Why are all of Mr. Shirley's gifts the same? I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers Does Process Builder run with System permissions or User permissions? Regarding field-level security, does Lightning Process Builder run as system-level permissions or user?  That is:   Can I trigger an update to a field that the current user cannot edit? Can I filter based on a field the current user cannot read? Windows 8.1 dont have hyper-v - cant use Windows Phone Emulator Unexpected voltage with switch turned off I am doing some work on a ceiling light and I notice that, even though the light-switch is off, my voltage tester shows voltage present.  Answers to other questions (below) explain about phantom voltages but how exactly do I test for these and know what voltages are real and what are phantom so that I can work safely?    Phantom voltage questions Prove no existing a smooth function satisfying ... related to Morse Theory Proper Way to Combine Feature Selection and Hyperparameter Tuning? I already have an Apple ID, made at Apple.com. Now, when I want to access the App Store from my iPad, I am told that I have to review my information. On the credit card page, I am not presented with a "None" option, even though I am in the United States.   What can I do to proceed without entering this information? Before upgrading to macOS Sierra, I really liked the Character Accent Menu by doing a long press on a key (for example, the character e) and being able to add an accent on it.   However, this functionality seems to not work anymore, and so far, I can't find out why?   Also, before, doing a long press on the delete button, would repeatly delete characters. Now, it is doing only one character deletion disregarding how long I'm pressing the key. So if I want to erase 10 characters, I need to press the delete button 10 times...  Also, I am not able to log off or shut down. When doing it, all applications are closed, status bar disappears but nothing happens. The dock remains here. I have no way to shut down or log out without doing a long press on the power button.  Is someone facing the same issue?  EDIT : Logging with a new user account or guest account fix all the problems. But I still would like to know what, in my user account, became corrupted during the upgrade. I want to draw a table and i used the following code  \begin{table}[htbp]  \centering \caption{Ship simulator required inputs} \begin{tabular}{|c|c|} \hline L_{PP},  L_{WL}, L_{OS}, B, T, T_{a}, T_{f}, C_{B}, D_{P}, `\\` Bulbous bow, Number of bosses, brackets, thrusters. &amp; ship particulars `\\` \hline Required speed or propeller rpm, ship course angle &amp; Mission data \\ \hline BN, weather angle, water temperature &amp; Environment \\  \hline \end{tabular}% \label{inputs}% \end{table}%`   the result of this is as shown in figure    How can i complete the right border of the table ??? I used \\ to break the line of first row Open a window on a remote X display (why "Cannot open display")? Once upon a time,  DISPLAY=:0.0 totem /path/to/movie.avi   after ssh 'ing into my desktop from my laptop would cause totem to play movie.avi on my desktop.  Now it gives the error:   No protocol specified Cannot open display:    I reinstalled Debian squeeze when it went stable on both computers, and I guess I broke the config.   I've googled on this, and cannot for the life of me figure out what I'm supposed to be doing.   (VLC has an HTTP interface that works, but it isn't as convenient as ssh.)  The same problem arises when I try to run this from a cron job. Game states & passing data between scenes I’m struggling with getting \ifmmode to work correctly when used inside an array environment which is inside a displaymath environment.  Here is a minimal example:  \documentclass{article}  \newcommand{\signal}[1]{%   \ifmmode     1#1   \else     2$#1$   \fi }  \begin{document} \begin{displaymath}   \begin{array}{cc}     \signal{a} \signal{b} &amp; \signal{c} \signal{d} \\   \end{array} \end{displaymath} \end{document}   This example generates the following output:        Why doesn’t all invokations of \signal detect math-mode? Homework question on whether two quotient spaces are homeomorphic Whatsapp, Keeping the number on the new phone but without the old sim card Whatsapp, Keeping the number on the new phone but without the old sim card. Ok the situation is I love my old number and to be honest don't want to ask everybody to change my number again and again. But I do not have the old sim card, and that bums me out. If there is any way to achieve this? I just installed Ubuntu 16.10 on my desktop and have a USB deathadder. The problem is I can't see where the cursor actually is because it appears only in the top left of my screen. The mouse still moves but the cursor graphic doesn't follow I accidentally closed the window for the verfication code I wanted to create an account for Ubuntu One, but I got impatient when the email with the verification code didn't arrive... I thought I might have typed the wrong address, closed the window and tried again. it told me the address already existed and I just got the code. But since I closed the window I missed it. Is there any way to recover the code? This is relating to the movie. Frodo finally relinquishes his desire for the ring, and it melts in the fires of Mount Doom. The next scene shows the eruption of Mount Doom and the destruction of Barad Dur. Gandalf looks up with tears in his eyes.  I have read the books, and can't seem to remember this being explained or described. I'm just wondering why they depicted Gandalf as being so emotional about the event, when throughout the story it depicts Gandalf as having unparalleled wisdom, faith and knowledge. Out of all the Ainur he listened to Eru's song the longest and contemplated it's meaning.   It strikes me that Tolkien portrayed Gandalf as all knowing, and has ultimate faith that Frodo would carry out the task at hand. So why was he so emotionally surprised that it happened? Especially when he had seen far more surprising things happening throughout the War of the Ring and in the movies he was depicted as emotionally immovable, an almost cold hearted awareness of the evil that lurks within the world.   I understand this is probably going to be down-voted as it has alot of opinion based statements. The TL:DR is: Was Gandalf described in the book as being so emotional at Sauron's fall? Or was this simply for effect in the screenplay? Are there any COGO tools in QGIS 1.9+? The "Azimuth and Distance" plugin does not appear to be maintained any longer, and has not been transitioned to the new QGIS 2.0 plugin repository. What are the current options for entering COGO in QGIS? Does the roadmap include these tools, or is this something that will still depend on a plugin?