Given that  $a&gt; 1$, $\alpha \in \mathbb{R} $,then  find $$\lim_{x\rightarrow \infty } \frac{a^x}{x^{\alpha}}$$ My attempts : $\frac{a^x}{x^{\alpha}} \le \frac{1}{x^\alpha}$ now  i take $\lim_{x\rightarrow \infty } \frac{1}{x^{\alpha}} = \frac{1}{\infty} = 0$ Is its correct ?? thanks  in advance
In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem.
How to prove that if A, B, C are submodules of M module then $$ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $$  Any help would be appreciated! Troubleshooting "Windows cannot be installed to this disk. The selected disk has an MBR partition table" I recently bought a desktop computer with Windows 7 Home Premium x64 preloaded (OEM). I tried to install Windows 7 Ultimate x64 on this computer but I was presented with the following error message:     Windows cannot be installed to this disk. The selected disk has an MBR   partition table. On EFI systems, Windows can only be installed to GPT   disks.   The 1TB HDD on this computer has three partitions:   SYSTEM 100 NTFS - Healthy (System, Active Primary Partition)  OS (C:) 9148.04 GB NTFS - Healthy (Boot, Page File, Crash Dump, Primary Partition)  HP_RECOVERY (D:) 13.37 GB NTFS - Healthy (PrimarynPartition)   I'm trying to install a clean copy of Windows. I'm having trouble distinguishing between the concepts of scedasticity and stationarity. As I understand them, heteroscedasticity is differing variabilities in sub-populations and non-stationarity is a changing mean/variance over time.   If this is a correct (albeit simplistic) understanding, is non-stationarity simply a specific case of heteroscedasticity across time? I have a complex integral $$\int_{|z|=r}x \, dz$$ for the positive portion of the circle.  I know that this integral seems easy enough but I am having trouble with it, and I'm fairly certain my answer is incorrect.  I wanted to use a parameter such that $x=r\cos\theta$, so we then have $dz=dx+i\,dy=dx$ since there is no reason to have have the $i\,dy$ since the imaginary part is equal to $0$.  So now I have the double integral  $$\int^r_0\int^\pi_0 r \cos\theta \, dx$$  which equals $0$ since integration the $\cos\theta$ gives $\sin\theta$ and evaluated from $0$ to $\pi$ makes the integral evaluate to $0$.  Am i misinterpreting part of the question?  Way off base?  The book suggests to also solve the problem by realizing $x=\frac{1}{2}\left(z+\frac{r^2}{z}\right)$ along the circle.  For this portion of the problem I assume that I would integrate the above by $z$ as if it were a normal calculus problem, but I have problems with what to put as upper and lower limits.  Thanks in advance! Aluminum foil is said to be not absorbing light at all. It reflects light. So, does it mean that a more shiny aluminum foil will reflect more light and thus make the room more cooler as compared to less shiny foil?  If yes, then where does unreflected light go in case of less shiny aluminum foil?  Does less shiny and more shiny actually make a difference? Why?  In order to keep whole room cool will it help if I stick the foil to the outside of wooden door (exposed to the sunlight) too? How to automate SSH login with password? How to automate SSH login with password? I'm configuring my test VM, so heavy security is not considered. SSH chosen for acceptable security with minimal configuration.  ex)  echo password | ssh id@server   This doesn't work.  I remember I did this with some tricks somebody guided me, but I can't remember now the trick I used... Twitter Bootstrap uses these break points for responsive design:  Large Display: 1200px Default: 980px Tablet: 768px Mobile: 480px  A client said they are using 1280px for their Large Display breakpoint.  Is there an industry standard for these or are they just made up as we go right now? How can I list the contents of a .zip folder in c#? How can I list the contents of a zipped folder in C#? For example how to know how many items are contained within a zipped folder, and what is their name? How to change the catcode in a macro How to connect to remote X-Server (logged in via ssh) When I'm logged on to another host (e.g via ssh), how do I connect to the XServer of that machine (same user is logged in and is running a desktop (gnome))?  You may ask way I wish to do that: There are commands that don't open an X-Window, e.g. xinput, xhost, etc.. and there are situations where you want to run them from remote. How to prove that exponential grows faster than polynomial? In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem. Getting text from the Resource file in asp.net 4 QuickStart basic layout/template for web development? I'm looking for a quick start or template, or something, to get a web site's basic layout done quickly and looking fairly professional, and then let me do the coding.  I don't want a CMS, as this is for a highly customised reporting application, but I need something like frame header and two columns and frame footer.  I can design the pretty reports for the wide right hand column myself, as the left hand column menu. Is there any way to make a method with more than one instance? If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis How to make tr aware of non-ascii(unicode) characters? Is there any way to add pasta to soup without the pasta doubling in size and getting mushy in the fridge later.I've even tried putting uncooked pasta in with the same result. Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square submatrix of $A$. It is true that the determinant rank is equal to the rank of a matrix, which we define to be the dimension of the row/column space.  It's not difficult to see that $\text{rank} \geq \text{determinant rank}$. If some submatrix of $A$ is invertible, then its columns/rows are linearly indepedent, which implies that the corresponding rows/columns of $A$ are also linearly indepedent.  Is there a nice proof for the converse? Press-and-hold or Long-Press Gestures: Unintuitive? Often, I find people struggle with the usage of press-and-hold gestures on mobile/web platforms. The mere thought of having a user press on a defined object and hold for an arbitrary amount of time before the product reacts seems overtly complicated and unobtrusive. My questions are:   For what use cases do press-and-hold gestures best fit? Have there been published studies with regards to press-and-hold gesture intuition? Are there more viable alternatives to this system? I would like to be able to set a flag at the beginning of the main flag (say output) that tells if I want to produce a paper document or an electronic document.   Then I should check this flag to define or not empty pages, hypperref, oneside or twosides, etc.  How could I do that ?