$\lim_{h\to 0} \frac1h\int_0^h \sin(1/t)\,dt$ Does$$\lim_{h\to 0} \frac1h\int_0^h \sin(1/t)\,dt\tag1$$   exist?    Let $F(h)=\int_0^h \sin(1/t)\,dt$. There is no closed form expression for $F$. If the limit in $(1)$ existed, then $F(h)$ would be an example of a differentiable function whose derivative is discontinuous, which is why I am interested. Solutions which are comprehensible by an exceptionally bright high school calculus student would be preferred, if possible.   My thoughts:   It is tempting to apply l'Hopital's rule to $\lim_{h\to 0}\frac{F(h)}{h}$, but $\lim_{h\to 0}\frac{F'(h)}{1}=\lim_{h\to 0}\sin(1/h)$ does not exist.  An equivalent expression, obtained from substituting $1/u$ for $x$, is $$ \lim_{h\to 0}\frac1h\int_{1/h}^{\infty}\frac{-\sin u}{u^2}\,du=\lim_{a\to\infty}a\cdot\int_a^\infty  \frac{-\sin u}{u^2}\,du.\tag2$$ It is tempting to bound $\left|\int_a^{\infty}-\sin u/u^2\,du\right|$ by $\int_a^{\infty}1/u^2\,du=1/a$, but this bound is too coarse. I strongly suspect the limit exists and is equal to zero. In $(2)$, I think the alternating sign of $\sin u$ causes $\int_a^\infty  \frac{-\sin u}{u^2}\,du$ to go to zero faster than $a$. However, I am getting tongue-tied in making this argument formal, especially using the tools available to a high-school calculus student (l'Hopital's, MVT, integration techniques, etc).   Thank you in advance for any help.
Is the function differentiable at $0$? Let $$f(x) = \begin{cases}\begin{align*}&amp;\cos{\dfrac{1}{x}}, &amp;x \neq0 \\ &amp;0, &amp;x=0. \end{align*}\end{cases}$$ Is the function $F(x) = \displaystyle \int_{0}^x f dx$ differentiable at $0$?  We can see that the function $f(x)$ is continuous everywhere except $x=0$. In order to show differentiability we will need to show the derivatives from the left and right are equal. So we need to show that $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h}.$$ The derivative from the right is $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^+} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}$$ and the derivative from the left is $$\lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}.$$ I am not sure how to proceed.
I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.  I tried applying the Mean Value theorem like this: There exists a $c \in (a, x)$ st $f(x)-f(a)=f'(c)(x-a) \implies \frac{f(x)-f(a)}{x-a}=f'(c) \implies f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{c\to a}f'(c)=A$  However, I think this is wrong. I am pretty sure the Mean Value theorem is needed, but my application of it seems to be incorrect. Would somebody be able to point me in the right direction? I have three displays, a 4K display flanked by two 1080p displays. Physically, these monitors are roughly the same size. Furthermore, Windows 10 handles the DPI scaling of the 4K monitor separately from the 1080p monitors just fine. So while it appears as such to Windows    the real layout is three monitors of equal size side by side.  The problem is that the mouse keeps getting caught on the edges of display 2 and three when dragging from display 1. I would like the mouse to continue onto display 2 or 3 regardless of which part of the left or right edge of display 1 the mouse touches.  This question has been asked before (e.g. ) and a common answer is to suggest that this behaviour is intentional on the basis that mouse movement should reflect different size displays. Such an answer is not appropriate as it doesn't consider DPI and scaling. Permission denied, are you root? Possible Duplicate:          On  it's a stated fact that if $I$ and $J$ are coprime ideals of a ring $A$, then $I^k$ and $J^\ell$ are also coprime for any integers $k,\ell$.   There's no proof given. Why is this fact true? There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process? How does quantum trapping with diamagnets work? I just saw  demonstration by someone from a Tel Aviv University lab.  What they achieved there is mind blowing. I myself own a levitron that uses the Hall effect to levitate a magnet, the problem with that is the magnet must always be flat facing the Hall effect base, any unbalance will wreck the levitation. But in the case of the video you can see that he isn't constrained by such things, the disk can levitate and stay still at whatever angle and orientation. Instead of being levitated, the disk is simply being held still.   How does this happen? He referred to this as quantum trapping, but I've little idea about what that is....  Also, what material is the base in these demos? Is the circular track in the demonstration magnetic? I assume this is so. Is there any conceivable way for this technology to make objects levitate on any surface? For $x,y\in I:=[0,1]$ define the relation on $I$ as $x-y\in \Bbb Q$. How big (using cardinal number) is the cardinality of the equivalence class $[1/\sqrt2]$?  I have tried to solve it by finding the equivalence class but I'm not sure about the method that I can solve the question. Please, can you help me to solve it. The Ball From A Pass If you hash a string using SHA-256 on your computer, and I hash the same string using SHA-256 on my computer, will we generate the same value? Does the algorithm depend on a seed (so we'd both need the same seed) or some other such parameter?  edit:  To clarify, by 'string' I meant 'the same byte input', which as the comments and @ispiro's answer point out, may be different for the same character string depending on the encoding. Chain-calling parent initialisers in python Notation for intervals I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed . I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.   What are the origins of the two notations?  Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this  Bourbaki is mentioned in connection with the notation $]a,b[$.)      Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this. The best I can come up with is the article on , where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.) I also tried to google for  or . Surprisingly, this lead me to a  where this notation is used for open interval. I know that it can be done with graphics packages like TikZ, but it seems to be a bit of an overkill. And for those not already familiar with the TikZ system, learning it can be a bit of a barrier. So my question is     Assuming the advanced graphics capabilities are not necessary, and I just need a bare-bones flow chart (which should be implementable just using the picture environment), is there a LaTeX package for simply making a flow chart? Baking lots of bread - keeping a yeast starter Like many people sheltering-at-home, I'm baking . My yeast supply is running low, and I'm not sure I'll be able to get more on demand. I'd like a simple way to make sure I have yeast on hand for my next batch. Could you all recommend a method for storing a starter culture?  Specifically:   How should I store it (how much? where? I have a fridge ~4C and a wine cooler ~13C)  How often should I feed it? With what? Should I divide the culture when feeding?  How much should I use when baking? Can I treat it like dry active yeast, or does it need more/less time?   For some other context, I'm a novice at making bread (I've made it a few times, and others in my family make it more often). I see lots of advice on sourdough starters, but am interested in something that's a replacement for dry active yeast (unless sourdough starters are actually easier to maintain). I found a  that was similar, but it focused on long-term storage of yeast and the answers did not describe how to use the starter.  Your advice is appreciated. We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions? Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime. Copy current terminal prompt to clipboard Is there a way to copy the currently written content of the bash prompt? Say I wrote:  ping www.google.com   so that the lines in terminal looks like:  petr@sova:~$ ping www.google.com   and didn't hit return yet. I want to copy that text to clipboard. Is that possible?  The usual usecase would be hitting up several times and finding a command which I want to copy. Alright Ubuntu wizards, I jacked up my system a bit unnecessarily, which is now giving me the chance to learn more about how to fix it...a vicious cycle. Anyway, I have it mostly up and running again to my liking, though I am receiving this (photo attached) prompt when my system boots:  libkmod: ERROR ../libkmod/libkmod-config.c:686 kmod_config_parse: /etc/modprobe.d/blacklist.conf line 56: ignoring bad line starting with 'rt2800usb'   Other users had success with just deleting the bad line from their blacklist.conf file, which I have done. Line 56 no longer exists, but I'm still receiving the same error. So am I missing something? Is there some follow-up command I need to give Ubuntu after saving the conf file? Is there a standard pronunciation for gender-neutral -@ or -x More and more I see, especially in activist communities, Spanish-derived words ending in x or @ in order to neuter the gendering inherent in the original language. For example: latinx (or latin@) as opposed to latina and latino. How, though, are these endings pronounced? Please provide evidence of how this orthography is most commonly pronounced. Note 1: I recognize this is in an English Language site, rather than a Spanish language site, but this phenomenon is very much happening in English, albeit with Spanish loan words. Therefore the question is relevant to non-Spanish-speaking English speakers. Note 2: A question like this can lead to politically charged or, at least, sarcastic comments. Please refrain from bringing in your biases for or against. Note 3: This question is not seeking opinions as to how some people pronounce the suffixes. Answers should address evidence of a developing orthodoxy in pronunciation. In my object record type i have created formula field. Now i want my formula field to work as below.  If Basement__c = NA and Property__r.Basement__c = NA OR  Basement__c = Crawl and Property__r.Basement__c = Crawl OR  Basement__c = Slab and Property__r.Basement__c = Slab Then it will return Type__c(formula field) = "MAtch"  And (Another rule) Basement__c DOES NOT contain any of the following "N/A,Crawl,Slab,None and Property__r.Basement__c Does not contain any of this then formula field will return "HAs basement"  How can i achieve this? Please help. Order of an element in a finite cyclic group Let $G$ be a cyclic group of order $m$ generated by an element $a$.  I want to show that the order of $a^k$ is $m/d$, where $d:=\gcd(k,m)$.  I have a simple proof, but want to make sure I haven't overlooked anything since the other proofs I've seen (such as in Dummit and Foote) look more complicated.    The order of $a^k$ is the cardinality of the set $\{a^{ks}: s \in \mathbb{Z}\}$, which equals the cardinality of the set $\{a^{ks} a^{mt}: s,t \in \mathbb{Z} \}$ since $a^m$ is the identity.  But the set $\{ks+mt: s,t \in \mathbb{Z} \}$ is the set $d \mathbb{Z}$ by the basic properties of integers. So we have that the cardinality of the set is $| \langle a^k \rangle| = |\{1, a^d, a^{2d}, \ldots, a^{m-d} \}| = m/d$.