Assume that $f: [0, \infty) \rightarrow \mathbb{R}$ is differentiable    for all $x&gt;0$ and $\lim_{x \rightarrow \infty} f'(x) = 0$. Prove that    $\lim_{x \rightarrow \infty}[f(x+1)-f(x)] = 0$   I was hinted that I should use the mean value theorem here. My attempt is as follows. Consider the closed interval $[x, x+1]$ where $x&gt;0$. Clearly, $f$ is continuous on $[x, x+1]$ and also differentiable on $(x, x+1)$ by the assumptions of the question. So we can apply the MVT and conclude that there exists a $c \in (x, x+1)$ such that $f(x+1) - f(x) = f'(c)$. Now if I take the limit to infinity on the left hand side, I can see the $\lim_{x \rightarrow \infty} f(x+1) - f(x)$ come into play, but what is $\lim_{x \rightarrow \infty} f'(c)$?   I thought about something like this, but not sure if it is right. Clearly, $c = x+t$ for some $0&lt;t&lt;1$, so $f'(c) = f'(x+t)$, so $\lim_{x \rightarrow \infty} f'(c) = \lim_{x \rightarrow \infty} f'(x+t)$. Now I am not sure how to bring $\lim_{x \rightarrow \infty}f'(x) = 0$ into the picture.
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.
Existence of the moment generating function and variance Can a distribution with finite mean and infinite variance have a moment generating function? What about a distribution with finite mean and finite variance but infinite higher moments? What is the reason Men play 5 sets & Women play 3 sets in a Grand Slam in Tennis? In the ATP &amp; WTA events, both Men &amp; Women tennis players play best of 3 sets. But when it comes to Grand Slams, the Men play 5 sets. At the risk of sounding a misogynist, I always believe the reason to be strength &amp; endurance. After all, a 5 set EPIC match lasts 5 hours while a 3 set EPIC last 3 hours.    Is that the only reason. Or am I a naive misogynist? Also was that the case since the history of Grand Slams or did Men play 3 sets before? In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$ Run script as root at login (no sudoer user, shell session) Much like /etc/profile and ~/.profile but ran by root instead of the user is doing login. /etc/rc.local runs after boot but I need running the script before login. The user is not a sudoer one.   Thanks! How do you set, clear, and toggle a single bit? In reading through the original GCM specification (), the composition of the 128 bit Initialization Vector as a concatenation of a 96b nonce and a 32b unsigned wrapping counter seems arbitrary and forces the scrambling pattern to repeat every 232 16-byte block.  Is the algorithm expected to be secure for significantly longer stream lengths if for example the IV were a 128b nonce XORed with a 64b or 128b counter, or are there known cryptanalysis issues that begin to arise? Arithmetic of continued fractions, does it exist? I'm interested in the arithmmetic of continued fractions and specially in multiplication. Consider  $$ f(x)=\cfrac{f_{0}(x)}{1-\cfrac{f_{1}(x)}{1+f_{1}(x)-\cfrac{f_{2}(x)}{1+f_{2}(x)-\cfrac{f_{3}(x)}{1+f_{3}(x)-\cdots}}}} $$   and   $$ g(x)=\cfrac{g_{0}(x)}{1-\cfrac{g_{1}(x)}{1+g_{1}(x)-\cfrac{g_{2}(x)}{1+g_{2}(x)-\cfrac{g_{3}(x)}{1+g_{3}(x)-\cdots}}}} $$   Are there arithmetic rules (algorithms) for the multiplication of continued fractions? Specifically, is it possible to obtain a continued fraction $h(x)=f(x)\cdot g(x)$ where   $$ h(x)=\cfrac{h_{0}(x)}{1-\cfrac{h_{1}(x)}{1+h_{1}(x)-\cfrac{h_{2}(x)}{1+h_{2}(x)-\cfrac{h_{3}(x)}{1+h_{3}(x)-\cdots}}}} $$   I've found , but I'd like more. Does anyone here knows of papers, algorithms, etc?  Thanks.    Update 1: Another . Did Luke attempt to retake the Jedi Temple on Coruscant? After the New Republic gains control of Coruscant through the Galactic Concordance, did Luke Skywalker attempt to reclaim (or even visit) the former and still very much intact Jedi Temple, now converted to be the deceased Emperor's palace?   If Luke is so insistent on saving a tree just because it belongs to the Jedi and the Empire stole it, then shouldn't reclaiming the Jedi Temple be even higher priority?  If he did, is he aware of the Sith shrine beneath it? Which is something even Yoda does not know. And if he didn't, why didn't he? How do you set, clear, and toggle a single bit? How do you set, clear, and toggle a bit? Can someone explain the layer system? I am really confused with the different uses of the terms "Layers" and "Passes" in Blender. Can someone please explain how they work in Blender. I know about Render Passes like AO, Diffuse and so on, but there seems to be a confusion with layers. Also, how do the layer buttons work in the properties panel like in the image shown? I get it that you can assign objects onto certain layers, but whats the difference between "Scene" and "Layer" buttons in the image? How do they refer to rendering? How can you see what layer a certain object is on? Let's say we are given a set of positive reals, and we're told that these are the edges of a convex  $n$-gon, and we must compute it's area.   For $n = 3$ there is the famous Heron's formula:  $$A = {1\over 4}\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}$$  For $n = 4$ there is Brahmagupta's formula:  $$A = {1\over 4}\sqrt{(a^2 + b^2 + c^2 + d^2)^2 + 8abcd - 2(a^4 + b^4 + c^4 + d^4)}$$  Is there a generalization for arbitrary $n$? Does it even exist? Change the passphrase of an ecryptfs directory For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere I want to show that for $f: \mathbb{R} \rightarrow\mathbb{R}$ Lebesgue integrable  $\displaystyle{\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere.  It is clear to me that this statement is true and I have tried to prove it by contradiction, by assuming that either $\displaystyle{\lim_{n \to +\infty} f(x+n) =\infty}$ or$\displaystyle{\lim_{n \to +\infty} f(x+n) =K}$ with $K\in \mathbb{R}$ .  In the first case  we can follow that $|f|=\infty$ so that $\int_\mathbb{R}f=\infty$ and we have a contradiction.  In the other case we know that there must be an $a\in \mathbb{R}$ so that $\forall x&gt;a$ we have $f_+(x)&gt;K/2$ and by that:  $\int_{{\mathbb{R}}+}|f|&gt;\int_{\mathbb{R}_+}f_+=\int_{[0,a]}f_++ \int_{(a,\infty)}f_+&gt;\int_{[0,a]}f_++\int_{(a,\infty)}K/2=\infty$  Would this be enough to show the statement? Can I even make the assumption that the limit of $f(x+n)$ "exists"? Any hints would be great. Compute the period of a decimal number a priori I noticed that , given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic decimal number, computes really fast the length of the period.  E.g. $\frac{3923}{6173}$ has a period of 3086: .  I was wondering how this computation is done: is there some method to do this (except the trivial one of executing the division and looking for a sequence repetition) ? Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$? Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$?   If I can show that $y^3+z^3-1$ is irreducible, I believe I can use Eisenstein's criterion. But I don't see how to show even this.  The other approach on my mind is to show that $k[x,y,z]/(x^3+y^3+z^3-1)$ is an integral domain, but this doesn't seem particularly promising either. How can I pay tolls on Japanese expressways? I'm renting a car soon in Japan and am planning to travel on the expressways. What's the best way to pay tolls? What is a word/phrase for using a term for a popular special case instead of a generic term? I am getting a warning when trying to include the .net 3.5 sp1 prerequisite for my setup project.  The warning states Prerequisite could not found for bootstrapping.  Any suggestions?  Thanks Will a PCI Express x16 3.0 card work in a PCI Express x16 slot? I have a motherboard with a PCIexpress x16 slot.  I want to purchase a card called AMD Radeon HD 7770, which uses PCIe version 3.0.  Is that backwards compatible? A term for adoption of a made up term Is there a term for the acceptance of a made up term that then became the name of the object/idea/action once it was invented?   Example: Cyberspace. It was first used by William Gibson in his cyberpunk novels, but when actually invented, it now is widely used across the world (or at least across the United States).  Is there a term for this phenomenon?