My Maple input limit(sin(1/n)*n,n=infinity); says 1.  I don't understand why $$ \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) \cdot n = 1 $$  I know that $\lim_{n \to \infty} 1/n = 0$, so it kind of says "0 * infinity = 1".  Have I overlooked some rewriting of $\sin(1/n) n$?
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.
How can I prove that if $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \times \gcd(b, c)$?  By eea there exists $ax+by=1$ from $\gcd(a,b)=1$ so a and be are co-primes there also exists $dk=a$ and $dj= b$ where $d=\gcd(a,b)=1$ this is all the information I have gathered from the question but I dont know how to approach and solve it. Can anyone help explain to me how to arrive at the answer? Thanks! Where to place my local website starting with the 2.4.7 version of apache2? I tried writing $y=x^{\sin x}$, so $\ln y=\sin x\ln x$. I tried to rewrite the function as $\ln(x)/\csc x$ and apply l'Hopital to the last function, but it's a mess.   Is there a by hand way to do it? Generating random numbers without repetitions What determines color -- wavelength or frequency? What determines the color of light -- is it the wavelength of the light or the frequency?  (i.e. If you put light through a medium other than air, in order to keep its color the same, which one would you need to keep constant: the wavelength or the frequency?) Why would Mystique's DNA allow for Sentinels that can duplicate mutant powers? The future Sentinels in  are clearly shown adapting to powers thrown at them by copying them. Isn't that what Rogue's DNA would allow for, not Mystique's? Mystique can just change appearance, not copy powers ... right? So why would study of her mutant genes allow for the abilities of the future Sentinels? Plot 4D data with color as 4th dimension Prove that $(p+q)^m \leq p^m+q^m$ If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$    Trial: For $m=0$, $(p+q)^0=1 &lt; 2= p^0+q^0$  and for $m=1$, $(p+q)^1=p+q =p^1+q^1$.   So, For $m=0,1$ the inequality is true.How I show that the inequality is also true for $0 &lt; m &lt; 1$.   Please help. I have created the custom setting with the hierarchy type    I have added some fields to the custom settings  phone field is added to the custom settings  custom setting Name:Order_Configuration__c  Phone :12345 - one profile  phone:456798 - another profile   Is it possible to access the different values for different profiles in custom settings using apex trigger code Why does this PNG image on twitter look different when clicked? This morning on twitter there was a tweet with the text "click on this picture".  When picture is clicked, it's opened in wider viewer, but looks completely different.  I tried to analyze this picture in ACDSee, and I also tried uploading on Facebook, but it's happening only when embedded on twitter. I don't have much knowledge about graphics so please explain this to me. Why does this PNG image change only when clicked on Twitter?  Screenshots (left = before clicking; right = after clicking):     Here's the original . Factorizing polynomials over fields other than $\mathbb{C}$ To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$ Let ${x_n}$ be a sequence in $R$ such that $|x_{n+1} - x_n|&lt; \frac{1}{n^2}$ for all $n \in N$.  Show that the sequence is convergent.  If it were $|x_{n+1} - x_n|= \frac{1}{n^2}$, could take help of the fact $\sum \frac{1}{n^2}$ is convergent and of triangular inequality.  But what to do here? Please help. Thanks in advance. Elements order in a "for (… in …)" loop Renaming accessor/mutator methods in Eclipse? Is there any way to automatically rename accessor/mutator when a variable they get/set gets refactored -> renamed (Eclipse 3.4)? Horizontal spacing issue in \listoftheorems using amsthm + thmtools Python — How can I find the square matrix of a lower triangular numpy matrix? (with a symmetrical upper triangle) Find and replace text within a file using commands Are all multiplicative functions additive? Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution function of $Z$. Can anyone give me some hints on how to proceed? I can only note that $\mathbb{P}[Z\geq 1]=1$ and $$F_Z(t)= \mathbb{P}[Z \leq t]=\mathbb{P}[X\leq Y, Y\leq tX]+\mathbb{P}[Y \leq X, x\leq tY]$$ CDN: How is it possible that my DNS delivers a different IP depending on the visitors location? I found  how a CDN works. But there is one thing I don't really understand. Let's assume I setup multiple DNS servers at my location and they use the nameserver domains dns1.example.com, dns2.example.com and dns3.example.com. This DNS servers are able to deliver a server IP depending on the visitors location (ping, geo database, browser language or whatever). Now I update this nameserver settings for my domain www.example.org at the registry.  Now, the very first request on www.example.org with an expired TTL tries to resolve the domain. It asks:   the local .hosts/DNS, if TTL expired: the internet providers DNS, if TTL expired: the root DNS, if TTL expired: my local dns1.example.com   But if I understand it correct, the new IP is then added to all these nameserver caches until the TTL expires again. So how is it possible to send other IPs to the visitor depending on his location?  In  theandym said the request is "forwarded", but I don't think this is how a CDN works, because "forwarding" means lengthen the transmission way resulting a longer loading time. Or does a CDN require zero TTL for the domain?  Update1 Through  I found  describing how they optimized CDN performance. It did not explain how the CDN works in general, but there were interesting explanations like the following:     Thereafter, whenever a client attempts to fetch content hosted on the   CDN, the client is redirected to the node determined to have the least   latency to its prefix. This redirection however is based on the prefix   corresponding to the IP address of the DNS nameserver that resolves   the URL of the content on the client’s behalf, which is typically   co-located with the client.   This means Google checks at first the latency of all IP prefixes and defines a DNS resolution table (?) for all available prefixes. And if a visitor has the IP 198.51.100.231 the Google server IP is used, that is set for the prefix 198.51.100.0. But again: How does Google's DNS know which IP the visitor is using? Most visitors resolve Google's domain through their internet provider and by that the resolving is done through those external DNS servers or not?  As an additional example: If I start a DNS resolution for the domain facebook.com with different online tools (hosted in different countries) it is resolved to different IPs with different domains like:   31.13.92.36 Reverse: edge-star-mini-shv-01-frt3.facebook.com 31.13.76.68 Reverse: edge-star-mini-shv-01-sea1.facebook.com 31.13.69.228 Reverse: edge-star-mini-shv-01-iad3.facebook.com 157.240.2.35 Reverse: edge-star-mini-shv-01-ort2.facebook.com   After that I thought it could depend on the DNS server location used by the visitor, but I tried my own (Deutsche Telekom, Germany), Google's (8.8.8.8) and a major one from France (Orange) and they all returned for facebook.com the IP 31.13.92.36.