I am currently taking a Survey of Calculus course and recently we started learning about these things called limits.  I know that they basically tell you what the "y" value of a function will be/be near, as "x" approaches a certain point, but I don't know what use this would be outside of finding "y" at certain points on a graph and derivatives of tangent lines (which becomes useless once you learn the "power rule.")  What exactly are limits, and what are some real life applications?  Are limits what Isaac Newton created to figure out where planets would be at certain points?
This morning, I read Wikipedia's informal definition of a limit:     Informally, a function f assigns an output $f(x)$ to every input $x$. The   function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever   $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$   as $x$ moves closer and closer to $p$.   To me that sounds like something that might be better described as a 'target'.  If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).   So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.   What is a simple example of a limit in the real world?  Thank you  -Hal.
Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$ How does Meta Stack Exchange work? Meta Stack Exchange seems to have different rules and user behavior than the "regular" Stack Exchange sites.  How is it different?    Individual questions answered below:                Related Initialization of an ArrayList in one line Fastest Gun in the West Problem Reading the question posted  left me with a more general question:  Given the professional title and name: Dr. (First Name)(Last Name), is there some way to differentiate between the holder of a philosophical doctorate and a medical doctor? Wouldn't it be more appropriate for a PhD holder to have the title (First Name)(Last Name), PhD? Inform users of certain tags on the existence of a new SE-2.0 site and offer to automatically post the question there instead Take for example  both at SO and SU. Now that there's , (new) users asking a LaTeX question should at least be informed that there is a new Stack Exchange site dedicated to TeX and thus increases their chance of an answer. This should happen before the question is actually posted and ideally offer the option to post the question directly there instead (without the need to copy-paste).  LaTeX is just an example, there's also , ,  etc., and it is obvious that this can only be decided on a case-by-case basis which is why I certainly do not suggest migrating questions automatically when these tags are used. But informing users of that possibility would probably help the SE network in general. Persist sort/filter selections the method getApplicationContext() is undefined less with "update file" like functionality I want to watch a file that gets overwritten every 5 minutes with less. How can I make less follow the new file descriptor instead of keeping the old one displayed? watch "cat file" won't do it because the file is too long to fit into one terminal window. What happens when you die? Diablo's first and second installments handled death differently, both in terms of exp\gold penalties and also in if you respawned or not or instead had to reload from an earlier point.  In Diablo III what happens when you die? Cooking away alcohol When you cook with wine or spirits, when does the alcohol cook away? Obviously high temperatures will do it, but how low of temperatures will work? Also, does it vary by the type of alcohol? Calculating the total number of surjective functions It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{m}}$, denoting the falling factorial). But I am thinking about how to calculate the total number of surjective functions $f\colon X \twoheadrightarrow Y $.  The way I thought of doing this is as follows: firstly, since all $n$ elements of the codomain $Y$ need to be mapped to, you choose any $n$ elements from the $m$ elements of the set $X$ to be mapped one-to-one with the $n$ elements of $Y$. This results in $n!$ possible pairings. But the number of ways of choosing $n$ elements from $m$ elements is $\frac{m!}{(m-n)!\,n!}$, so the total number of ways of matching $n$ elements in $X$ to be one-to-one with the $n$ elements of $Y$ is $\frac{m!}{(m-n)!\,n!} \times n! = \frac{m!}{(m-n)!}$.  Now we have 'covered' the codomain $Y$ with $n$ elements from $X$, the remaining unpaired $m-n$ elements from $X$ can be mapped to any of the elements of $Y$, so there are $n^{m-n}$ ways of doing this. Therefore I think that the total number of surjective functions should be $\frac{m!}{(m-n)!} \, n^{m-n}$.  Is this anything like correct or have I made a major mistake here? Why are there no other dragons in ASOIAF? Finding determinant for a matrix with one value on the diagonal and another everywhere else Let us look the the matrix  $\left(\begin{array}{ccccc} a &amp; b &amp; b &amp; b &amp; b\\ b &amp; a &amp; b &amp; b &amp; b\\ b &amp; b &amp; a &amp; b &amp; b\\ b &amp; b &amp; b &amp; a &amp; b\\ b &amp; b &amp; b &amp; b &amp; a \end{array}\right)$  It has one value, $a$, on the main diagonal, and another value, $b$ everywhere else. Let us assume that we are over a ring and that $a$ is invertible.  I wish to find the determinant of every $n\times n$ matrix of this form ($a$ on the diagonal, $b$ everywhere else). Using row and column operations I have managed to transform the matrix to upper-triangular form and found formula for specific cases. Generalizing it I got to the following formula:  $\det(A) = a\left(a-b\right)^{n-2}\left(a+\left(n-2\right)b-\frac{\left(n-1\right)b^{2}}{a}\right)$  I think I can prove it with row-operations in the general case with a little patience.  However, I'm wondering if there is a "smart" way of getting to this formula that I'm missing and if there is a nicer representation of it. Also, what can be said when $a$ is not invertible? (esp. the case where we are over a field and $a=0$). Today I updated to version 2.18.52 &amp; I can’t see the names of the people who send me a message. The only thing I see is the WhatsApp logo and the word “message”.    I can’t also see the status my friends posts, they all look black. I uninstalled the app, reset my phone &amp; nothing seems to work so far. Labels disappear while zooming Proof of conservation of energy? How is it proved to be always true? It's a fundamental principle in Physics based on all of our currents observations of multiple systems in the universe. Is it always true to all systems? Because we haven't tested or observed them all. Would it be possible to discover/create a system that could lead to a different result? How are we 100% sure that energy is always conserved? Finally, why did we conclude it's always conserved? What if a system keeps doing work over and over and over with time? Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$ If $x&gt;0$, $y&gt;0$, $z&gt;0$ and $xyz = 1$ then   $$\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$$   I tried using $\displaystyle x = \frac{a}{b},y = \frac{b}{c}$ and $\displaystyle z = \frac{c}{a}$ substitution,  $\displaystyle \sum_{cyc} \frac{\sqrt{a^2+b^2}}{b} \le \sqrt{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right) \\ \displaystyle  \iff 6+ 2\sum_{cyc}\frac{\sqrt{(a^2+b^2)(b^2+c^2)}}{bc} \le \sum_{cyc}\frac{a^2+b^2}{b^2} + 4\sum_{cyc}\frac{b}{a}$  So it would suffice if we showed $\displaystyle \sum_{cyc}\frac{\sqrt{(a^2+b^2)(b^2+c^2)}}{bc} \le 2\sum_{cyc}\frac{b}{a}$  Squaring again does not lead anywhere nice. Is the last inequality true ? If not is there a better way of showing the result ? If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite.  so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this point, and also $f(x)$ must be continuous at x as well  so that mean that:  $\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} =  \lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$  and that: $\lim_{x\to x_0^+}=\lim_{x\to x_0^-}=f(x_0)$  I know I should probably use arithmetic limit laws to prove this but I can't see how what I figured out could help me. any help with that?  Thanks! Is there any difference between "offense" and "offence"?