I'm a math student going over some old exam questions. I have arrived at a number of questions that ask me to compute the limit of a function near a given number (2 in this case). But they appear to be harder than I first realised.  $$\frac{x^2 - 4}{x-2} = \frac{1- \frac{4}{x^2}}{\frac{1}{x} - \frac{2}{x^2}} \iff \lim_{x \to 2} \frac{1- \frac{4}{x^2}}{\frac{1}{x} - \frac{2}{x^2}}= \frac{1-1}{\frac{1}{2} - \frac{1}{2}} = \frac{0}{0} $$  Even though I used the algebra of limits I still arrived at a division of zero. I feel I have made a rookie mistake or assumption, but if anyone can help me fill the hole in my logic that would be much appreciated!
I have passed the calc sequence, and thought that I understood things.  But I have trouble with limits now and again.  From the way I understand it, the 'best' way to evaluate a limit, is to evaluate the expression at the number in the limit.  In that way it's not only a limit, but an algebraic expression (so, I think, the need for limits goes away entirely, superseded by accuracy.)  But then I see something like $\lim_{x\to 2} \frac{x^2-4}{x-2}$.  Now, I think in my mind 'the best way to evaluate this is just to plug in $2$'  But it doesn't work that way, as the limit is $4$, and if I plug in, the answer is $0/0$.  I think that people here, with their natural talent in math, can perceive where my train of thought is going wrong, and correct me.  Please!
Question on determinants of matrices changing between integer matrices The following problem came up from a though I had while reading:  Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$.  We know we can make a matrix $A$ that changes $N$ to its representation in $M$ (ie has as columns the basis $N$ expressed in coordinates coming from the basis of $M$). If the index of $N$ in $M$ is $m$, I am pretty sure just from tinkering that the determinant of $A$ should also be $m$, however, I have not been able to show this.  It is unclear to me how to proceed. I though maybe I could do something related to the points in $\mathbb{Z}^n$ that are within the unit box of $N$, but I cannot really make sense of this.  Thank you for any help or direction. Is it normal for focal length to change while focusing? When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change.  More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance.  This is most noticeable at the 55mm end of the lens.  Is this normal? How long does the Truffle take to move in after building the mushroom biome? I currently find myself in a pickle because even after I've built the mushroom biome and met all other requirements, the Truffle NPC won't move in.   I've defeated not only the Wall of Flesh but Ocram already. Created a surface mushroom biome. Built a house made out of mushrooms with mushroom furniture.   I have no idea how long it takes for the NPC to move in or if I'm wrong in any of the steps. I want to calculate the derivative of $|x|^\alpha$ with respect to $x$, where $1&lt;\alpha&lt;2$.  $\frac{d|x|^\alpha}{dx}=\alpha|x|^{\alpha-1}\mathrm{sign}(x)$  Is it correct?  And is it true that  $\lim_{x \to 0} \alpha|x|^\alpha\mathrm{sign}(x)=0$?  What with the following derivative $|x|^\alpha\rm{sign}(x)$. I would be grateful for any hints. Can polymorph and true polymorph change a PC's race and sex? What's using up my disk space? My Vista 64 installation has drive C, a 150 GB hard drive.  I also has a 1.5 TB drive D.  My C drive has 12.3 GB free.  I don't know what's using up all the space.  There's almost 83 GB unaccounted for!!!  As per  question and  question, I downloaded  and .  WinDirStat, once I found the right options, tells me there's 82.9 GB of .  SpaceMonger, which has a particularly nice display of the data, tells me there's 82.3 GB "Unscanned", with 2 unscannable folders.  Where do I turn now?  Both tools roughly agree there's a large amount of data on my drive, but neither seem to be able to tell me what.  I've tried the standard approach; disk cleanup, remove hibernation file, set the virtual memory to the D drive instead of the C drive. What does ` (backquote/backtick) mean in commands? I came across the following command:  sudo chown `id -u` /somedir   and I wonder: what is the meaning of the ` symbol. I noticed for instance that while the command above works well, the one below does not:  sudo chown 'id -u' /somedir Word for person who looks for flattery What word could be used to describe a person who seeks and seems to depend on constant positive feedback from the people around him. What methods can I use to create balance and consistency between a group of differing logos? A regular challenge that I face is balancing two (or more) logos. Whether they need to be equal or one needs to be given priority. Some logos however might be squares, others rectangles, circles, etc...   As a quick sample I snagged a nice high resolution photo from  and pretend its going to be an advertisement that needs to show some footwear brand logos on it. In this case I used Adidas and Saucony.  This is certainly not balanced even though they're height is the same:    Likewise this is not balanced even though they're width is the same:    So then the question becomes when are they in balance, is this balance:    Maybe a little more adjustment, is this balance:    And in these examples I'm only working with two Black Logos. Factor in color and it can get even more difficult to balance. Another issue being when there's tiny text as part of the logo.  So the question is, other than estimating / eye-balling it, what tips, methods or even tools are there to balance different sized logos? Irreducible polynomial is always separable in char 0 field Way to go from recursion to iteration I've used recursion quite a lot on my many years of programming to solve simple problems, but I'm fully aware that sometimes you need iteration due to memory/speed problems.  So, sometime in the very far past I went to try and find if there existed any "pattern" or text-book way of transforming a common recursion approach to iteration and found nothing. Or at least nothing that I can remember it would help.   Are there general rules? Is there a "pattern"? How to remove white space before reference? New messages flagged blue bubble does not go away When someone flags a message in chat all 10k+ users in chat get a little blue bubble to vote on. It shows up in the upper left hand corner of your gravatar (which is in the bottom left of the page).  When someone flags a message and it shows up, if I wait for a second to click on it, and then click on it, and it has already been handled, it shows the message  "There are no flags to display."  In the past, after showing this message, the blue (1) would disappear. However, lately it has not been going away and requires a page refresh in order to remove. Can this be fixed?  This issue is addressed here:  from September 12th 2012. But since it is still occurring 10 months later, I believe the cause is different. Sequence of rationals with an irrational limit have denominators going to infinity What are complex numbers, actually? I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)  My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?  For me personally it doesn't matter if I write $\cos(360°)$ or $\cos(2\pi)$. Both equals 1, so why bother with two conventions? The error I am having when trying to update my package list.  W: GPG error: http://download.opensuse.org ./ Release: The following signatures were invalid: KEYEXPIRED 1436387333  W: Failed to fetch http://ppa.launchpad.net/samrog131/ppa/ubuntu/dists/trusty/main/binary-amd64/Packages  404  Not Found  E: Some index files failed to download. They have been ignored, or old ones used instead. What was L's real name? I watched through Death Note once, and I never remember hearing L's name, even after he died.  What was L's real name, or did he never have one (it was never said)? Printing pointers in C Disk storage graphic and usage amount inconsistent?