So i'm a little shaky on limits and I want to show  I could take the limit of$  \frac{x^{2.5}}{100^{x}} $  but it would get very messy with L'Hopitals rule is their an easier way to show that this goes to 0 without having to do  a bunch of algebra?
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.
Can't install php5 on Ubuntu 16.04 so I've recently installed my first Ubuntu, to be precise, a daily build of 16.04, on my new laptop since the hardware was to new for 15.10. Now I'm in the process of setting it up and I want to install php5 and some extensions.  When I execute sudo apt-get install libapache2-mod-php5 I get the error message that there is no installation package available for php5 and libapache2.  The error message is in german, so I doubt that this would help, but anyways here it is. :-) E: For package »libapache2-mod-php5« existiert kein Installationskandidat.  Do I need to add a apt-get repository? Or what am I doing wrong? Centre of Lie Algebra $sl_2(\mathbb{F})$ For $L=sl_2(\mathbb{F})$ i.e. matrices with trace zero,  what is the centre i.e. $Z(L)$= {$x\in L : [x,y]=[y,x] \ \forall\ y \in L$}.  I will have to find matrices $A \in L$ such that $AB=BA$ for all $B \in L$. But how to appoach this considering trace is zero here. Prove that $\frac{ 5^{125}-1}{ 5^{25}-1}$ is a composite number How to play MP3 files using Rhythmbox? Do I need Intel Smart Response when installing Ubuntu? Does registering [name]overflow.com violate Stack Overflow's trademark? When a land becomes creature, can you kill it? I tried many ways to make a bigger screen in VirtualBox because I do not like the small size of the guest window? Is there is a way to make it big? Computing determinant without expansion $$\begin{align}\mathrm D &amp;= \left|\begin{matrix} (b+c)^2 &amp; a^2 &amp; a^2 \\ b^2 &amp; (a+c)^2 &amp; b^2 \\ c^2 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)\left|\begin{matrix} b+c - a &amp; a^2 &amp; a^2 \\ b - a -c &amp; (a+c)^2 &amp; b^2 \\ 0 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right| \\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ b - a -c &amp; a+c - b &amp; b^2 \\ 0 &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ 0 &amp; a+c - b &amp; b^2 \\ c - a-b &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\end{align}$$  Can $\rm D$ be further simplified without expanding ?  I feel it should be because this was competition question. Why does adding a curve modifier move my object to the center of the curve? I'm using a curve modifier and an array modifier. My object is at one end of the curve but as soon as I choose the curve for the curve modifier it moves to the center. How can I fix the problem so that I can have the array going from one end of the curve to the other? Why is it better to use "#!/usr/bin/env NAME" instead of "#!/path/to/NAME" as my shebang? I notice that some scripts which I have acquired from others have the shebang #!/path/to/NAME while others (using the same tool, NAME) have the shebang #!/usr/bin/env NAME.  Both seem to work properly.  In tutorials (on Python, for example), there seems to be a suggestion that the latter shebang is better.  But, I don't quite understand why this is so.  I realize that, in order to use the latter shebang, NAME must be in the PATH whereas the first shebang does not have this restriction.  Also, it appears (to me) that the first would be the better shebang, since it specifies precisely where NAME is located.  So, in this case, if there are multiple versions of NAME (e.g., /usr/bin/NAME, /usr/local/bin/NAME), the first case specifies which to use.  My question is why is the first shebang preferred to the second one? Notation: Why write the differential first? System freezes/unresponsive/unusable when copying large file to USB Stack Exchange Android app keeps replaying notifications Version of the app: 1.0.51. (It is up to date at the time of writing according to Google Play.)  This started happening after the network problem that happened earlier this morning. Both my tablet and my phone are popping up notifications for events (comments, and probably edits too) that happened prior to the network issue. I had received notifications for these events already and had already cleared them before the network issue happened.  The repeated notifications do not show up in the bar displayed at the top of the Stack Exchange sites. In other words, I get notified again in Android but do not get notified again on the SE web sites. When is R squared negative? My understanding is that $R^2$ cannot be negative as it is the square of R. However I ran a simple linear regression in SPSS with a single independent variable and a dependent variable. My SPSS output give me a negative value for $R^2$. If I was to calculate this by hand from R then $R^2$ would be  positive. What has SPSS done to calculate this as negative?   R=-.395 R squared =-.156 B (un-standardized)=-1261.611   Code I've used:  DATASET ACTIVATE DataSet1.  REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA             /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN             /DEPENDENT valueP /METHOD=ENTER ageP   I get a negative value. Can anyone explain what this means? I am using XeLaTeX and memoir to write a longer text with several parts, chapters and sections. When printing the \tableofcontents I get a spacing problem with 2-figure numbers almost overlapping the relevant section name, as per enclosed picture.  I am aware that the memoir manual has a section dedicated to this theme, but the most part of it is doubledutch to me. Is there any easy way to avoid this problem? Model won't follow its armature in pose mode I've set up a rig and built it around my model, and I've linked it to my model but when I rotate the bones in pose mode, the character doesn't move with them.   My rig is the human meta rig that comes with rigify. I've seen tons of videos of it working. Why isn't it working on mine?  My character was made by creating different body parts and joining them together into the same object, then a texture was added by unwrapping them in texture mode. Keeping the time stamp when syncing with Ubuntu One? Smooth option sometimes produces incorrect arrow tips in PGFplots PS4 Controller not connecting to PS4 and won't reset? I was playing my PS4 two days ago and then packed it up with two controllers that were both connected to this PS4.   I got to where I was going, set everything up and now the controllers won't connect. They show pulsing orange when they plug in, but when I hit the PS Button to connect, it blinks white 1-2 times and goes back to orange.   I have tried resetting them both.  Can anyone help?