Let $f:[0,\infty)\to\mathbb{R}$ be bounded on every interval $[0,b]$ for $b\gt0$. Assume that $$\lim_{x\rightarrow\infty}\left(f(x+1)-f(x)\right)=0.$$ How do I prove that $$\lim_{x\rightarrow\infty} \frac{f(x)}{x}=0?$$
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 $\lim\limits_{x\to+\infty}\dfrac{f(x)}x = L$
I don't think this has yet been covered in .  There was , however, it was not specifically talking about the case where the proper noun ends in a plural noun.  Feel free to vote to close if I am wrong.  What is the correct way to make a proper noun ending in a plural noun ending in an "s" possessive?  This frequently happens with corporations, e.g., "Dunkin' Donuts."  Should one work off of the fact that the entity is singular—suggesting Dunkin' Donuts's—or should one work off of the fact that "Donuts" is plural and ending in an "s"—suggesting Dunkin' Donuts'?  I expect that the answer might depend on dialect because some regions refer to corporations in the singular form ("Dunkin' Donuts is a company") while other regions refer to corporations in the plural form ("Dunkin' Donuts are a company").  I am specifically interested in American English, but would be interested in hearing answers for other dialects too. I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. Permutations with repeating digits My question is this : how many distinct two digit numbers can be produced from numbers $4, 3, 3, 1$? When applying the formula $$\frac{4!}{(4-2)!2!}$$ you come up with $6$, yet when doing the problem manually, I come up with $7$ numbers, namely $41, 43, 14, 13, 33, 31, 34$. Can anybody explain this discrepancy?  Thanks. I installed Redhat 6 x86_64. I am using the Network connection screen to set a static IP address like below (I want two PC's in my house to see each other: one Redhat PC and one Mac)   192.168.0.5   255.255.255.0   192.168.0.1     When I run ifconfig it displays only lo and virbr0 information. I don't know what these items are (I don't really know much about network settings).   When I try ifconfig -a it displays eth0, lo, sit0 and virbr0. The information for  eth0 is as follows:   Link encap : Ethernet HWaddr 90:2B:34:74:05:30 BROADCAST MULTICAST MTU:1500 Metric:1 RX packets:192 errors:0 dropped:0 overruns:0 frame:0 TX packets:6 errors:0 dropped:0 overruns:0 frame:0 collisions:0 txqueuelen:1000 RX bytes 53811 (52.5 KiB) TX bytes:468 (468.0 b) Interrupt:29 Base address:0xc000   Could someone help me to point out if anything wrong with my setting or how to resolve this problem? I just uninstalled TeX Live 2016 and installed TeX Live 2017 and I noticed that the vertical alignment of author and title in any beamer presentation with Warsaw theme has been disrupted. In this example the author is JESTIN and the title is Beamer Intro. How can I vertical center them?  Moreover, I just found the same bug in the beamer guide obtained writing texdoc beamer in the command prompt, in the part of a warsaw presentation.    Here is my code:  \documentclass{beamer}  \usetheme{warsaw}  \title[Beamer Intro]{Beamer - Introduction to \LaTeX{} presentation making} \author[JESTIN]{Jestin Joy}  \begin{document}  \begin{frame}   \titlepage \end{frame}  \end{document} Regarding the ellipsoid computed by using the "lifting step" I need your expertise in understanding and hopefully solving the following problem:  Let $C \subset \mathbb{R}^d$, and lets assume that $C$ is not centrally symmetric, thus when computing the $MVEE(C)$ (i.e the minimum volume enclosing ellipsoid), they use the following "lifting step":  $$ C^\prime = \left\lbrace \pm \begin{bmatrix} x\\ 1 \end{bmatrix} \middle| x \in C \right\rbrace,$$ where $C^\prime$ is centrally symmetric.  It is proven that for any $\varepsilon &gt; 0$, $MVEE(C^\prime) \cap \Pi$ is a $(1 + \varepsilon)$-approximation to $MVEE(C) \times \left\lbrace 1 \right\rbrace$, where $\Pi = \left\lbrace x \in \mathbb{R^{d+1}} \middle| x_{d+1} = 1 \right\rbrace$.  Let $G \in \mathbb{d \times d}$ be an invertible matrix, let $ v \in \mathbb{R}^d$ such that the pair $(G,v)$ represents the matrix and the center of $MVEE(C)$ respectively. Let $\hat{G} \in \mathbb{R}^{(d + 1) \times (d + 1)}$ be an invertible matrix such that the pair $(\hat{G},\overset{\to}{0})$ represents the matrix and center of $MVEE(C^\prime)$ respectively.  Then, $\forall x \in MVEE(C)$ $$ \left| \left| \left[ x , 1 \right] \hat{G} \begin{bmatrix} x\\1 \end{bmatrix} \right| \right|_2 = (1 + \varepsilon) \cdot \left| \left| (x - v) G (x - v)^T \right| \right|_2 .$$  Regarding $x \not\in MVEE(C) $, then it is easy to see that the above term is not satisfied and moreover the right term is bigger or equal to the left term.  What I need to know is whether it possible the bound the right term by the left term (maybe because $MVEE(C) \times \left\lbrace 1 \right\rbrace \subseteq MVEE(C^\prime)$), i.e. $$ poly(d) \cdot \left| \left| \left[ x , 1 \right] \hat{G} \begin{bmatrix} x\\1 \end{bmatrix} \right| \right|_2 \geq (1 + \varepsilon) \cdot \left| \left| (x - v) G (x - v)^T \right| \right|_2 \ ?$$  Please Do correct if I am wrong and thanks in advance. When is it a good idea to let an academic department (or a professor) know that I have Asperger's (or ADD)? With Asperger's, the trick is always when to explain it (in a non-awkward situation). I'd like to avoid mentioning it if possible, but it has gotten to the point that I only mention it after committing a major social faux pas, in which case it might be seen as an excuse.  With ADD, this is an issue that happens when people explain things to me. I often don't parse things very well when they're communicated to me verbally since I do have attention lapses. That's the major issue, anyways.   With that said, I do (indirectly) imply that I have both diagnoses on my personal website. Question on determinants of matrices changing between integer matrices The following problem came up from a though I had while reading:  Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$.  We know we can make a matrix $A$ that changes $N$ to its representation in $M$ (ie has as columns the basis $N$ expressed in coordinates coming from the basis of $M$). If the index of $N$ in $M$ is $m$, I am pretty sure just from tinkering that the determinant of $A$ should also be $m$, however, I have not been able to show this.  It is unclear to me how to proceed. I though maybe I could do something related to the points in $\mathbb{Z}^n$ that are within the unit box of $N$, but I cannot really make sense of this.  Thank you for any help or direction. Which is correct: "what if there was" or "what if there were"? Possible Duplicate:          Is this correct grammar?     What if there was a Stack Overflow on…    Shouldn't it be "what if there were a Stack Overflow on…"? If $x+{1\over x} = r $ then what is $x^3+{1\over x^3}$? If $$x+{1\over x} = r $$ then what is $$x^3+{1\over x^3}$$  Options: $(a) 3,$ $(b) 3r,$ $(c)r,$ $(d) 0$ Can I have a mathematical equation within a caption? I'd like to have an eqnarray environment (or equivalent) within a table caption. If I do this naively,  \begin{table*}   % table stuff   \caption{\label{tab:my_tab}     \begin{eqnarray}       a &amp;=&amp; b \\ c &amp;=&amp; d     \end{eqnarray}   } \end{table*}   LaTeX falls over with LaTeX Error: \begin{table*} ended by \end{eqnarray}. How can I get what I want (I don't mind no equation numbers)? Lie algebra and tangent space at the identity are isomorphic Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success. Additive Roman Numeral 4 I am working on a diplomatic edition of a 15th century book and it numbers its folios using the standard roman numerals of the time.  I've been using the following to get the page/folio in my header:  \newcommand{\rectoverso}[1]{\ifthenelse{\isodd{#1}}{r}{v}} \newcommand{\folio}[1]{\the\numexpr (#1+1)/2\relax} \uppercase\expandafter{\romannumeral \value{\folio{\thepage}}}\rectoverso{\thepage}   This works great, but the problem is that the roman numeral sequence is off of what I need.  Instead of I, II, III, IIII, V, VI, VII, VIII, IX, X, I get the subtractive IV (somewhat understandably, given it's the far more common one in modern usage).  My source text only (and always) uses the additive 4 in the one's position: 24 is XXIIII, 34 is XXXIIII but 44 is XLIIII.  9 is universally rendered subtractively as IX or XC.  Is there a package that allows for some control over which numbers are done additively and which are subtractively, or would my best option be to write my own custom definition to account for the specific style used in my book? Probability of continuous random variable Arbitrary products of quasi-coherent sheaves? Title is the question itself.  How can I show below without L’Hospital’s Rule  $$ \lim_{x\to \infty} ( 1 + x )^\frac{1}{x} = 1 $$ How can I rename many files at once? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. How to show only date after the date of today in JCalendar test class with Batch and Callout I have a problem with FeedItems in my Test class, it's very weird behaviour Does someone explain to me why this happens?  Callout Class  public with sharing class Callout {  public static void runProccess(Case[] records) {     Database.executeBatch(new CalloutBatch(records), 10); }  public static void send(Case[] records) {     Case[] toUpdate = new Case[]{};       for (case record : records) {         HttpRequest req = new HttpRequest();         req.setEndpoint('callout:test/SOME_URL');         req.setMethod('POST');         req.setHeader('Content-Type', 'application/json');         req.setBody('');         Http http = new Http();         HTTPResponse res = http.send(req);          Map&lt;String, Object&gt; response = (Map&lt;String, Object&gt;) JSON.deserializeUntyped(res.getBody());          toUpdate.add(new Case(Id = (Id) response.get('Id'), Status = (String) response.get('Status')));     }      update toUpdate;  } }   CalloutBatch Class  global with sharing class CalloutBatch implements Database.Batchable&lt;Case&gt;, Database.AllowsCallouts{  private final Case[] records;  global CalloutBatch (Case[] records) {     this.records = records; }  global Iterable&lt;Case&gt; start(Database.BatchableContext BC) {     return records; }  global void execute(Database.BatchableContext BC, Case[] scope) {     Callout.send(scope); }  global void finish(Database.BatchableContext BC) { } }   Test Class  @IsTest  private class CalloutTest { @IsTest static void shouldReaciveData() {     String status = 'Closed';     Case caseRecord = new Case(Type = 'Web', Status = 'New');     insert caseRecord;      //insert new FeedItem(ParentId = caseRecord.Id, Visibility='AllUsers', Type = 'TextPost', Body = '');      System.assertEquals(0, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]);      Test.startTest();     Test.setMock(HttpCalloutMock.class, new CalloutTestMock(caseRecord.Id, status));     Callout.runProccess(new Case[]{caseRecord});     Test.stopTest();      System.assertEquals(1, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]); } }   Mock Class  public with sharing class CalloutTestMock implements HttpCalloutMock {  private Id recordId; private String status;  public CalloutTestMock(Id recordId, String status) {     this.recordId = recordId;     this.status = status; }  public HTTPResponse respond(HTTPRequest req) {     HttpResponse response = new HttpResponse();      response.setStatusCode(200);     response.setBody('{"Id": "' + recordId + '", "Status": "' + status + '"}');      return response; } }   This code works, but when I uncomment insert FeedItem in Test Class I receive an error message:     System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out   Class.Callout.send: line 23, column 1   Class.CalloutBatch.execute: line 20, column 1   FeedItem doesn't have an impact on this functionality, (I don't have any Process Builders, triggers etc. on FeedItem object) Best choice between using \\ or leaving space after each paragraph to end the paragraph Here is the thing, I have always used a blank space between paragraphs in order to tell latex that there is a new paragraph. But there is someone I work with very often that keeps using \\ to break a line after a paragraph...  This is really annoying so I wanted to know if there is a real advantage of using \\ over just leaving a space.  Note that I am not referring to the paragraph environment, but to the simple term paragraph.