It's been awhile since I last touched Calculus so I'm asking for help in a really basic question. I'm trying to prove the following theorem:     Let $a_n$ and $b_n$ be sequences so $\lim_{n\to\infty} a_n\cdot b_n=1$ and $\lim_{n\to\infty} b_n=\infty$. So we get $\lim_{n\to\infty} a_n=0$.   I'm trying to prove this theorem by the limit definition but I can't seem to figure out how to find $\epsilon$. Is it possible to show how?
A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$  B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=0$  C. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n= \infty$    These seem to be difficult situations I come across when I have to evaluate limits.   Some examples,    $\lim_{n\to\infty}n^a\sin(1/n^b)$ with $a,b&gt;1$   $\lim_{n\to\infty}{n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty}{1/n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty} nf(x_n)$ [where we only know that $\lim \ f(x_n) = 0]$    My first question is : Are the two situations  A and B  equivalent ? I feel that they are equivalent as $\lim_{n\to\infty}b_n=\infty \iff \lim_{n\to\infty}{1\over b_n}=0$, The problem is that I'm not sure if we can apply the product rule for limits to take the limit inside.  My second question is : What are some ways to solve such problems ? I only know that we can apply stolz- cesaro in some cases.
Solving 2-degree equations in 3 variables. We are given 3 equations: $x^2+\sqrt3 xy + y^2 = 25$ $y^2 + z^2 = 9$ $x^2 +xz+ z^2 = 16$. $x,y,z$ are positive real numbers. Then we have to find value of $xy + 2yz + \sqrt3 xz$. How do I read the manifest file for a webapp running in apache tomcat? PSTricks error "Undefined control sequence  \c@lor@to@ps" I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. Would I be scrutinized at the border for traveling to the US twice in a quick succession? I am a British citizen visiting the USA in early August for a two week vacation with friends. After the holiday I'm returning to my home in the UK for three weeks. After that I will return to the USA and will be working for my British employer under the Visa Waiver Program for a period of 80-85 days.  I have read already that there is technically no problems with this under ESTA rules, but should I limit this working period to a shorter time frame (e.g. 70 or so days) so that it does not negatively affect the decision made by the POE officer? Ideally I would love to be over in the states for the near-full 85 day period.  Any advice or previous experiences anyone can offer?  Some clarifications: As confirmed by the immigration attorney, as I am employed by my UK employer, it will be regarded as a business trip, the important part being that I will be working officially as a consultant for my UK employer just overseas for the allotted 85 or so days How can I find the value of the sum $\sum_{k=1}^{\infty}\frac{k^n}{k!}$? for example for $n=6$, we have $$\sum_{k=1}^{\infty}\frac{k^6}{k!}=203e.$$ Manually installing a program on Debian X forwarding from Ubuntu Server Ed to Windows 7 The Goal: I would like to open a program from my ubuntu server in my windows 7 desktop.  Steps Taken: Putty > SSH > X11 check 'enable port forwarding' I have started VNC Viewer in listening mode started putty ssh session apt get install xinit startx &amp; geany (it's a c language ide; it uses the GUI so I"m using it to test x forwarding)    Error Messages: geany returns "cannot open display" startx returns "setversion 1.4 failed"    I'm working on Windows 7 and headless Ubuntu Server 12.04 LTS  Obviously I forgot something, but I'm having difficulty figuring out what. The goal is to not use sudo apt get install desktop, since I only want to use one program at a time, and anyway my server isn't hooked up to a monitor.   Any ideas? Questions? Thanks for your help! Curios Does standard c library provides linked list etc. data structures? How to convert image of map into vector format? I have read enough threads on QQplots here to understand that a QQplot can be more informative than other normality tests. However, I am inexperienced with interpreting QQplots. I googled a lot; I found a lot of graphs of non-normal QQplots, but no clear rules on how to interpret them, other than what it seems to be comparison with know distributions plus "gut feeling".   I would like to know if you have (or you know of) any rule of thumb to help you decide for non-normality.  This question came up when I saw these two graphs:    I understand that the decision of non-normality depends on the data and what I want to do with them; however, my question is: generally, when do the observed departures from the straight line constitute enough evidence to make unreasonable the approximation of normality?  For what it's worth, the Shapiro-Wilk test failed to reject the hypothesis of non-normality in both cases. I read that this is because the mean and variance completely specify a normal distribution. However, e.g. for a Poisson distribution the mean, variance and third moment can all be expressed with the same parameter lambda. Is this not what it means to 'completely specify' a distribution?  What is an example of two Poisson distributions having the same mean, variance and different higher moments? If we treated weak stationary as if it was strict stationarity, what would happen when we try to do time series analysis? Is there a name for "the string of characters typed randomly on a keyboard home row"? Calculator Program improvements I'm pretty new to programming and made a few programs so far. I recently made a Calculator program and I hope you guys could tell me what I could do to improve. int firstNumber; string sign; int secondNumber; int answer;  Console.WriteLine(&quot;Input Number:&quot;); firstNumber = Convert.ToInt32(Console.ReadLine()); Console.Clear();  Console.WriteLine(&quot;Input Sign:&quot;); sign = Console.ReadLine(); Console.Clear();  Console.WriteLine(&quot;Input Second Number:&quot;); secondNumber = Convert.ToInt32(Console.ReadLine()); Console.Clear();  switch (sign) {     case &quot;+&quot;:         answer = firstNumber + secondNumber;         Console.WriteLine(answer);         Console.ReadKey();     break;     case &quot;-&quot;:         answer = firstNumber - secondNumber;         Console.WriteLine(answer);         Console.ReadKey();     break;     case &quot;*&quot;:         answer = firstNumber * secondNumber;         Console.WriteLine(answer);         Console.ReadKey();     break;     case &quot;/&quot;:         answer = firstNumber / secondNumber;         Console.WriteLine(answer);         Console.ReadKey();     break;     default:         Console.WriteLine(&quot;Invalid Input&quot;);     break; } Form inside a table Gauge fields and strings: Loop equations Find the cardinality of the quotient of $\mathbb R$ in respect to R R is an equivalence relation defined as $xRy \Leftrightarrow a - b$ is an integer.  What is the cardinality of the quotient of $\mathbb R$ in respect to R? How would you prove it?  I thought about a bijection between $[0,1)$ to the quotient of $\mathbb R$ in respect to R, but can't figure out to prove it formally. LaTeX Editors/IDEs Why, historically, do we multiply matrices as we do? Multiplication of matrices &mdash; taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product &mdash; is not a very intuitive operation: if you were to ask someone how to mutliply two matrices, he probably would not think of that method. Of course, it turns out to be very useful: matrix multiplication is precisely the operation that represents composition of transformations. But it's not intuitive. So my question is where it came from. Who thought of multiplying matrices in that way, and why? (Was it perhaps multiplication of a matrix and a vector first? If so, who thought of multiplying them in that way, and why?) My question is intact no matter whether matrix multiplication was done this way only after it was used as representation of composition of transformations, or whether, on the contrary, matrix multiplication came first. (Again, I'm not asking about the utility of multiplying matrices as we do: this is clear to me. I'm asking a question about history.) Not only in ortho: top, front, back.  Please don't link this answer   I want a background image like this guy has -> I need to clip a vector layer to the extent of a raster. It has to be very precise that's why I would not want to draw a polygon myself (and clip the two polygons). There seem to be some ways to clip a raster to a polygon but I have difficulties to find a solution to clip a polygon to a raster. I already tried to create a polygon out of my raster several times in ArcGIS and R, so I could clip the result. But every time it crashes because the raster is to big. There must be a simple way to solve this, and somehow I can just not see it at the moment.  How can I do this using GDAL?