Prove that if $\displaystyle \lim_{x \to \infty} f(x)$ and $\displaystyle \lim_{x \to \infty} f'(x)$ are both real numbers, then $\displaystyle \lim_{x \to \infty} f'(x) = 0$.   Attempt  Intuitively this makes sense to me. Take $y = x$. This slope is constant but it increases arbitrarily, and it seems that we can't make both the slope and value of $f(x)$ to be real numbers without "flattening" out the graph. I tried first saying $\displaystyle \lim_{x \to \infty} f(x) = a$ and $\displaystyle \lim_{x \to \infty} f'(x) = b$. Then we might be able to do something with the L'Hospital's rule.
I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks
What is the most secure hand cipher? By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.  Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.  For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.  So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex? Let's say that you want to output or concat strings. Which of the following styles do you prefer?   var p = new { FirstName = "Bill", LastName = "Gates" }; Console.WriteLine("{0} {1}", p.FirstName, p.LastName); Console.WriteLine(p.FirstName + " " + p.LastName);   Do you rather use format or do you simply concat strings? What is your favorite? Is one of these hurting your eyes?  Do you have any rational arguments to use one and not the other?  I'd go for the second one. Can one PC throw another? How do you convert a matrix to a cell array of equivalent size? Can someone pgfplots the following picture concerning confidence intervals? I've been trying to use the following code to produce it, but have no idea how to shade in the tails I am interested in. By the way, the line pointing the (100-C/2)% can be an arrow, does not have to be squiggly as depicted in the picture.   \begin{tikzpicture}[ every pin edge/.style={&lt;-}, every pin/.style={fill=yellow!50,rectangle,rounded corners=3pt,font=\small}] \begin{axis}[every axis plot post/.append style={   mark=none,domain=-3:3,samples=50,smooth}, clip=false, axis y line=none, axis x line*=bottom, ymin=0, xtick=\empty, ] \addplot {\gauss{0}{0.5}}; \addplot {\gauss{0}{1}}; \node[pin=70:{$\hat{\theta}_1$}] at (axis cs:0.57,0.5) {}; \node[pin=270:{$E(\hat{\theta} = 0$}] at (axis cs:0,0) {}; \draw[dashed] (axis description cs:0.5,0) -- (axis description cs:0.5,0.92); \end{axis} \end{tikzpicture} I'm trying to match SHA1's in generic text with a regular expression.  Ideally I want to avoid matching words.   It's safe to say that full SHA1's have a distinctive pattern (they're long and a consistent length) - so I can match these reliably - but what about abbreviated SHA1's?  Can I rely on the presence of numbers?  Looking at the SHA1's in my commit log - numbers always appear in the first 3 characters. But is this too short? How many characters of SHA1 do I need to consider before I can assume a number would have appeared?    This does not have to be 100% accurate - I just need to match an abbreviated SHA1 99% of the time. How to model an entity type that can have different sets of attributes? IS or ARE? "The only thing that I want you to hit right now IS/ARE the books" Missing dollar problem This sounds silly but I saw this and I couldn't figure it out so I thought you could help. The below is what I saw.  You see a top you want to buy for $\$97$, but you don't have any money so you borrow $\$50$ from your mom and $\$50$ from your dad.  You buy the top and have $\$3$ change, you give your mom $\$1$,your dad $\$1$, and keep $\$1$ for yourself.  You now owe your mom $\$49$ and your dad $\$49$.  $\$49 + \$49 = \$98$ and you kept $\$1$.   Where is the missing $\$1$? Run cURL commands from Windows console Is there a way to install cURL in windows in order to run cURL commands from the command prompt? Do not enumerate a section, but add it into table of contents How do I rid of number of a section in the way that it still would be in the table of contents.  If I use   ... \section{SN1} ...   then I see this section name in table of contents as   1 SN1  2 SN2  3 SN3  ...   If I use   \section*{SN}   I don't see it in the content:   1 SN2  2 SN3   How to get in content just   SN1  1 SN2  2 SN3   ? I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that.  The author defines an embedding between two manifolds $f: N\rightarrow M$ as an injective immersion (differential is injective everywhere) with the additional property that $f(N)$ as a subspace of $M$ is homeomorphic via $f$ to $N$. Then a canonical form theorem is proven, namely, that for any point $p \in N$, there are charts $(U,\phi = x^1, \ldots, x^n) \ni p$ and $(V,\psi = y^1, \ldots, y^m) \ni f(p)$, such that the transition map takes the following form:  $$\psi \circ f \circ \phi^{-1} : (r^1,\ldots,r^n) \mapsto (r^1,\ldots,r^n,0,\ldots,0).$$  (I guess it is implicitly assumed that $f(U) \subset V$).  Now in the process of proving that $f(N) \subset M$ is an , he states that $f(U) \subset V$ is defined by the vanishing of the coordinates $y^{n+1},\ldots,y^m$. I can prove that $$ f(U) \subset \{q \in V: y^{n+1}(q) = \cdots = y^m(q)\} $$ but not the reverse inclusion. Can anyone explain why this should be true? Name for these kinds of "sayings"? Suppose $X$ and $Y$ are independent positive random variables with probability density functions $f_X$ and $f_Y$ respectively. Show that $Z=X/Y$ is absolutely continuous and find its probability density function.   $\textbf{My Thought:}$ In order to show that $Z$ is absolutely continuous, I need to show that it has a commulative distribution function. Then I can also differentiate the latter to obtain the probability density function of $Z$. I have the following calculation \begin{align*} P(X/Y\leq z)&amp;=P(X\ge zY,Y&lt;0)+P(X\le zY,Y&gt;0)\\ &amp;=\int_{-\infty}^{0}\left(\int_{yz}^{\infty}f_{X}(x)dx\right)f_{Y}(y)dy+\int_{0}^{\infty}\left(\int_{-\infty}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy\\ &amp;=\int_{0}^{\infty}\left(\int_{0}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy. \end{align*} Also, if $z&lt;0$, then we have that $F_Z(z)=0,$ since $X,Y$ are positive random variables. Therefore the random variable $Z$ is continuous, and we can differentiate it to obtain the probability distribution function of $Z$. We have that $f_Z(z)=0$ if $z\leq 0,$ and otherwise we have $$f_Z(z)=\int_{0}^{\infty}yf_{X}(yz)f_Y(y)dy.$$    Is my reasoning above correct?  Any feedback is much appreciated.  Thank you for your time. Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is similar to $B$. Prove $\operatorname{trace}(A) = \operatorname{trace}(B)$.  I'm not sure where to go on this. So far I have this:  If $A$ is similar to $B$, then      $B=P^{-1}AP$ and $A=PBP^{-1}$   This implies that:     $\operatorname{trace}(B) = \operatorname{trace}(P^{-1}AP)$ and $\operatorname{trace}(A) = \operatorname{trace}(PBP^{-1})$   Not sure where to go from here whether it be vs whether it is I brought the sentences from the speech about conversation tips.     They don’t just want to know about news and the weather.   They want to know about you.   So what are three things that you have going on right now, that you can talk about.   Whether it be school, whether it be a personal relationship, whether it be a hobby that you’re involved with.   My questions:     What is the difference in nuance between 'whether it is' and 'whether it be'?   Before posting this question, I googled, and some websites say 'whether it be' is subjunctive. I don't get why it is called 'subjunctive'. Because from what I know about subjunctive, it's supposed to be past of be verb.    If I say 'whether it should be' or whether it would be', is the meaning same? Open maps which are not continuous What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to $\mathbb{C}$ (filled in to be $0$ at $0$). There must be a simpler example, using the usual Euclidean topology, right? If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? Is it possible to double your proficiency bonus twice? Let's say I'm a multiclass rogue/ranger and I have taken Expertise in survival checks. Now, if a ranger is in their favored terrain it says they double their proficiency bonus in intelligence and wisdom checks relating to you environment. If I am in my favored terrain and use my already doubled survival check, will it double again? The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file After executing this command  sudo apt-get update &amp;&amp; sudo apt-get -y upgrade   I get this error  W: The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file. N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use. N: See apt-secure(8) manpage for repository creation and user configuration details.   I've seen past inquiries on same issue but the answers given are specific to other repositories.  Any help will be appreciated.