$\lim\limits_{x \to 0} \frac{\tan(mx)}{\tan(nx)} =$?
$\lim_{x \to 0} f(x)=\frac{\tan 3x}{\tan 2x}$ without L'Hospital
Access a custom .NET DLL in VBScript I wrote a DLL in .NET and I want to access it in VBScript. I don't want to add it to the assembly directory.   Is there a way to point too the DLL and create an instance of it? Accidentally moved directories under root I have two partitions, one of them has a file system and the other is unallocated.  How to merge two partitions? How can I change wallpaper automatically? I have found a guide for that but doesn't sure if it will work or it is safe to use. The guide:  I am asking if is these introductions safe and will work? If not, is there any other solutions for that? I am using Ubuntu 20.04 LTS Thank you. Proving a distance between molecules defines a metric space. A DNA molecule can be represented as a string of symbols $A$, $C$, $G$ and $T$, such as  $$GGATAATTCTAG. . .GACCGTACCC$$ For the purposes of this question, we will assume that all DNA molecules contain the same (large!) number $N$ of symbols. Thus, a DNA molecule is an $N$-tuple $x = (x_1, . . . , x_N)$ where $x_i  ∈ \{A,C,G,T\}$ for each $i$.  Define the distance between two DNA molecules $x$, $y$ as the number of elements $i ∈ {1, . . . ,N}$ such that $x_i \neq y_i$. Prove that this defines a metric on the set of DNA molecules. Local colormaps in global coordinate system I am trying to get a surface graph with two colormaps.  Instead of using the full Z range i want to give the first surface its own colormap  and the second surface should receive its own colormap too (it may be even the same)... however the current version takes the full range of Z-Data and calculates the colormap. Is there any way?  \begin{tikzpicture}     \begin{axis}[grid = major, colormap={bw}{gray(0cm)=(0); gray(1cm)=(1)}, view={-30}{15}]   \addplot3[surf]    table     {DATA1.dat};   \addplot3[surf]    table     {DATA2.dat};  \end{axis} \end{tikzpicture}   Right now it looks like that:   I want each of the surface to be scaled individually... just to see more clearly how the amplitude changes :)  Any ideas?  Thanks, Alex Is there a way to offset an edge? (extrude edge loop perpendicular to each edge) I recently figured out how to do it with faces, but with edges, I can't seem to get it to work. I have a complex flat plane and i need to extend an edge loop. I tried setting to individual origins in combination with all possibilities for x,y,z axes in global and local, but it is still not extruding along the normals.   actual problem   simplified Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"? interaction between mathematical structures From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one... What is the difference between encrypting something and hashing something?  in what situations would I want one or the other? Best way to start investing, for a young person just starting their career? I am young, just started my career, and looking to begin investing so I can make my money work for me. I'm still living like I did in college (occasional purchase on myself, but most of it goes into savings) and I would like to take the extra money and invest it.   What would you suggest are the best options for me to begin? I was considering getting a personal investor [a financial advisor] for perhaps a year or so, but I've heard that they really are not worth the money. I wanted to get the input of the great minds here. Terminal only shows up as a black window ls: cannot access .gvfs: Permission denied Is "89.9%" singular or plural? I’m writing a dissertation on network security. I’m unsure which of the following to use:   ...89.9%, which is 57 million users.  ...89.9%, which are 57 million users.   Technically it should be are because it refers to a plural amount of people, but it doesn’t sound right to me considering that 57 million is in a sense one group. How to change keyboard layout via command line cmd.exe on Windows XP/7? Is is possible to change the keyboard layout via command line on Windows XP/7? What do you call a person who "shuts down" ideas? I noticed that $f(x) = x\phi(x)$ seems to be one-to-one, where $\phi(x)$ is Euler's Phi function.  In particular, I'm writing some numerical python code and the line I have looks something like  sorted([n*phi(n) for n in range(1,1000)])   and there are no duplicates in the list.  First, is it one-to-one?  Second, if it is, is there a simple proof sketch? Fermat's Last Theorem for Gaussian Integers (excluding the integers or  same pure imaginaries) I am investigating solutions to Fermat's equation $$x^n + y^n = z^n$$ with $x,y,z$ in the Gaussian integers, excluding integers and pure imaginaries.  I have found out that there are only trivial solutions for the $n=3$ and $n=4$ cases, e.g. .   I would be grateful if you let me know of the current status or if it is already a theorem. Recent reports claim that the gravitational waves detected by LIGO match up with the signal expected from two black holes merging as predicted by general relativity.  Additionally, the masses of both black holes were estimated. Questions:  How certain are we that the detected of gravitational waves are necessarily from two black holes merging?  Since supermassive black holes are typically at the center of a host galaxy, what happened to the galaxies that contained the two merging black holes?  For example, did their host galaxies merge?    Edit After reading the answers, I understand that the signals detected by LIGO are proof of the validity of gravitational wave generational artifacts as predicted by general relativity.  When additionally gravitational waves, caused by the merger of the supermassive black holes (those which ARE indeed located in the centers of the merging galaxies) will be detected, since mergers of galaxies could be observed by the methods independent of gravitational waves such as registering the fact of the quasar appearance (). Additional questions:  Could someone offer me the historical ASTRONOMY precedent when the second degree of inference (from the mathematical model, being associated with the theory of Physics) was accepted as the discovery of the astronomical object?  Could someone give a reference to trustable scientific publication, with the subject of study being discussion of requirements satisfying claim of astronomical discovery of an astronomical object?  Have independent scientific sources (outside of members of the LIGO team) analyzed methodology and results of the LIGO signal detection, and published their conclusions with regards of what was actually discovered? Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these  and saw the quaternion group, but the subgroups of the quaternion group aren't abelian...