Why does simplifying a function give it another limit I'm asked:   $$\lim_{x\to 1} \frac{x^3 - 1}{x^2 + 2x -3}$$  This does obviously not evaluate since the denominator equals $0$. The solution is to:   $$\lim_{x\to 1} \frac{(x-1)(x^2+x+1)}{(x-1)(x+3)}$$ $$\lim_{x\to 1} \frac{x^2 + x + 1}{x + 3}$$ $$\frac{1+1+1}{1+3} = \frac{3}{4}$$  My question: what is actually happening? How can simplifying a function give it another limit? Is it a complete other function and if so why would it be relevant to our original question?
Why does factoring eliminate a hole in the limit? $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much.
Prove that $||x|-|y||\le |x-y|$ I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much. What is the relationship between symmetry and degeneracy in quantum mechanics? Access is denied because the NTFS volume is already exclusively opened What Happens When A Gravitational Wave Interacts With Another One? If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one. Lock your computer after 5 minutes of inactivity How to solve and see resolution of $13^{53} \pmod 7$ using Fermat little Theorem? How can I wake-up a PC using a command? (i.e. equivalent to moving a mouse, etc) It doesn't seem like this would be very difficult, but how can I wake-up a PC by doing the equivalent of moving a mouse or pressing keys on a keyboard?  My PC is running Windows 10 and I only have the monitors enter sleep mode after 15 minutes.  The rest of the PC is still awake and doesn't use hibernate, etc.  I'm trying to use EventGhost to receive a TCP/IP command that tells the PC to wake up the displays but it's proving quite difficult. Normal ubuntu desktop will update ubuntu-desktop package and it's dependencies on release upgrade. Will Ubuntu minimal cd install ubuntu-desktop on release upgrade or just installed packages? Code is supposed to be exact and "as-is". Sometimes, showing two spaces is important. In a comment, it is not possible to show code with  multiple   spaces. This became apparent in  (see comments).  EDIT: apparently this bug is larger then I thought: multiple    spaces   here are neither preserved in normal question text. Pity (though I couldn't find this reported before, I'd assume it is).  Workaround: use Alt+0160 (windows), or U+00A0 (Unicode NBSP), as I did in the example above. I forgot my username, how do I get it back? Divisibility by 7 rule, and Congruence Arithmetic Laws I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = (a_ka_{k-1}\ldots a_2a_1a_0)_{10} = \displaystyle{\sum_{j=0}^{k}}a_{k-j}10^{k-j}$. The expression $$ Q_{3}^{\prime}(n) = (a_2a_1a_0)_{10} - (a_5a_4a_3)_{10} + (a_8a_7a_6)_{10} -\ldots  $$ are called alternating sum of the digits of third order of $n$. For example,  $$ Q_{3}^{\prime}(123456789) = 789-456+123=456 $$ Proposition: $7 | n \ \Leftrightarrow \ 7 | Q_{3}^{\prime}(n)$.  proof. ??  Thanks for any help. Reading a bit about recent Monica Cellio related posts, I am afraid that I may be using a language that is a lot like English, but might not be the Stack-Exchange-approved version of English. For example, my English has had a certain gender-neutral pronoun in living use for centuries. That pronoun is used throughout the literature written in this version of English. It is used in the English I was taught as a grammar school student and later as a graduate student working on a master's thesis. I believe that this English is good enough to communicate about topics relevant to the various Stack Exchange sites.  The last thing I want from Stack Exchange is to be booted out because I slip up and accidentally use the language that has served me so well for a lifetime.  The topics discussed on Stack Overflow (I almost referred to it as a "sister site;" would that have been allowed? "mother site?" Good thing I didn't?) are often centered on languages which have formal descriptions.  Where is the formal description of the language that is to be used on Stack Exchange?     Strive not with a man without cause, if he have done thee no harm.      He shall die without instruction; and in the greatness of his folly he shall go astray. What's the difference between declaring a variable "id" and "NSObject *"? In Objective-C, what's the difference between declaring a variable id versus declaring it NSObject *? Would a wormhole be able to move? How to compute   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$   My Working :   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} =  1-0 = 1$$  Is it correct What is the C# Using block and why should I use it? What is the purpose of the Using block in C#? How is it different from a local variable? Open Mapping Theorem: counterexample The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to have some counterexamples. After all, how can you appreciate it's meaning without a nice counterexample showing how the conclusion could fail and why the conclusion is not obvious at all.  Let $\ell^1 \subset \mathbb{R}^\infty$ be the set of sequences $(a_1, a_2, \dotsc)$, such that $\sum |a_j| &lt; \infty$. If we consider the $\ell^1$ norm $\|\cdot\|_1$ and the supremum norm $\|\cdot\|_s$, then, $(\ell^1, \|\cdot\|_1)$ is complete, while $(\ell^1, \|\cdot\|_s)$ is not complete.  In this case, the identity $$   \begin{array}{rrcl}     \mathrm{id}:&amp; (\ell^1, \|\cdot\|_1)&amp; \to &amp;(\ell^1, \|\cdot\|_s)     \\                 &amp; x &amp; \mapsto &amp; x   \end{array} $$ is a continuous bijection but it is not open.  I want a counterexample in the opposite direction. That is, I want a linear continuous bijection $T: E \to F$ between normed spaces $E$ and $F$ such that $F$ is Banach but $T$ is not open. This is equivalent to finding a vector space $E$ with non-equivalent norms $\|\cdot\|_c$ and $\|\cdot\|_n$, such that $E$ is complete when considered the norm $\|\cdot\|_c$, and such that $$   \|\cdot\|_c   \leq   \|\cdot\|_n. $$ The Open Mapping Theorem implies that $\|\cdot\|_n$ is not complete.  So, is anyone aware of such a counterexample? Why do we need virtual functions in C++? Today I bought an ibook G4 from a garage sale. The people who sold it to me didn't know the password for the 2 accounts on it. Apparently it was the guys late father's laptop. Any suggestions on how to get into the laptop and set a new admin account and password? Since canonical is  Ubuntu One, what are the possible alternatives?  I currently use Ubuntu One as a backup service for my Documents folder and various other folders I deem important. Of course I have backups on an external hard drive as well, but nevertheless, I find Ubuntu One to be very useful. What I'm especially looking for is a cloud service that does not need a special folder (such as a dropbox or google drive folder); something that is able to integrate into my own file system and syncs just any folder I want to.   I am aware that this question is rather subjective, but I do guess many people who have used Ubuntu One are now in the dark.