I tried to solve this for finding limits, $$\lim_{x\to\infty} \frac{\sin^2(\sqrt{x+1}-\sqrt{x})}{1-\cos^2(\frac{1}{x})}$$ step 2 by applying conjugate $$\lim_{x\to\infty} \frac{\sin^2\left((\sqrt{x+1}-\sqrt{x})*\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}\right)}{1-\cos^2(\frac{1}{x})} = \lim_{x\to\infty} \frac{\sin^2\left(\frac{1}{\sqrt{x+1}+\sqrt{x}}\right)}{1-\cos^2(\frac{1}{x})}$$ and applying the basic trignometric rule, $$sin^2(\theta)+cos^2(\theta)=1$$ $$\lim_{x\to\infty} \frac{sin^2(\frac{1}{\sqrt(x+1)+\sqrt(x)})}{sin^2(\frac{1}{x})}$$ Here if I apply the limits I will get $$\frac{0}{0} $$form. Is there another way of simplification for the above step. The referred link for the same question haven't got better results. They to arrive at the solution $$\frac{\infty}{\infty}$$.  Thanks.
How to evaluate the limit of this expression?  $$\lim_{x\to\infty} \frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$$  I managed to simplify the denominator into a sinus form by the Pythagorean formula, and also modified the argument of the sinus in the numerator by dividing with a conjugate which I think is necessary, into this final form:  $$\lim_{x\to\infty} \frac{\sin^2\left(\frac{1}{\sqrt{x+1}+\sqrt{x}}\right)}{\sin^2\frac{1}{x}}$$  And that's where I got stuck. So far I've tried to come up with some solution to eliminate the sinus from the denominator, without success. I'm sure the solution is pretty obvious but I don't quite know how to modify the trigonometric functions in this fashion, so any help would be most appreciated.
Evaluate : $1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$ Evaluate : $$1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$$  thought: I have tried to manipulate the series so that the sum upto n terms , can be simplified. But I can't figure out any profitable result. How to synchronize clocks for computers that are not online? I want to reinstall Ubuntu from scratch.   I have read ( and ) that it is recommended to have 3 partitions for the desktop installation: root, home and swap as having a separate home and root can simplify the process if you want to upgrade or reinstall Ubuntu (if you break something by mistake).  But there is no type of formula (that I can find) on how big each partition should be. I know this is somewhat dependant on what you plan to install. I want to have flexibility down the line to install new software without having to worry about shifting the partition sizes around.  Does anyone know of a formula or percentage breakdown that they have used to successfully calculate partition sizes?  i.e. x% of space will be allocated to the swap, y% of space to the root partition and z% of space to the home partition Prove that if $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m &lt;n.$   Since $G$ is of prime-power order I know $|Z(G)| \ne e$ so there is an $a\in Z(G)$ with order $p$ such that $p \mid |Z(G)|$. Now, the subgroup generated by is normal since it's a subgroup of the center. How can I get this normal subgroup to be less than $n$? Powers of infinite cardinals in the generic extension Very strange error suddenly appeared running Biber I am on a Mac with MacTex 2020 and ran into a strange problem with Biber. Working on a document all was fine, I did not change the bibliography (says my git) and only fixing typos (not even adding cites) and suddenly latexmk stopped working after a Biber run, while that looked like  INFO - This is Biber 2.14 INFO - Logfile is 'doc.blg' INFO - Reading 'doc.bcf' INFO - Found 31 citekeys in bib section 0 INFO - Processing section 0 INFO - Globbing data source 'literature.bib' INFO - Globbed data source 'literature.bib' to literature.bib INFO - Looking for bibtex format file 'literature.bib' for section 0 INFO - LaTeX decoding ... INFO - Found BibTeX data source 'literature.bib'   so harmless in my view. Then with --debug I additionally get in the end.  INFO - LaTeX decoding ... warning: overriding existing definition of macro "jun" warning: overriding existing definition of macro "jul" warning: overriding existing definition of macro "sep" warning: overriding existing definition of macro "jan" [...] warning: overriding existing definition of macro "nov" INFO - Found BibTeX data source 'literature.bib' Unicode::UCD: failed to find unicore/version in /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc CODE(0x7fd026aeb5a8) CODE(0x7fd026aebb90) at /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib/Biber/Input/file/bibtex.pm line 1011.   It seems unrelated to the library itself since deleting a few entries (going through deleting one half, then the other), the problem persists. It does not happen for an empty literature, but as soon as a first entry is added (just author/journal/title/year/volume/pages - no utf8-only chars) this error message appears.  what does this last error message mean?  How can I solve this? How far does a rogue move when dashing twice using Cunning Action? A rogue receives a bonus Cunning Action, and one option is to Dash. My understanding is a Dash allows a player to move up to twice his speed for his turn.  If a rogue is using a regular action to Dash (hence, is already dashing) can he also use his cunning action as a bonus to Dash again — effectively moving 4 times normal speed in a turn? I'm working through Knuth's Metafont Book as preparation for learning Metapost and I'm running into some problems working through his "experiments".  I've installed the latest TeX Live 2013 distribution on my system but I'm having problems.  Here's the code (contained in file io.mf as recommended by Knuth):  mode_setup;  em#:=10pt#; cap#:=7pt#;  thin#:=1/3pt#; thick#:=5/6pt#;  o#:=1/5pt#; define_pixels(em,cap); define_blacker_pixels(thin,thick); define_corrected_pixels(o);  curve_sidebar=round 1/18em; beginchar("O", 0.8em#, cap#, 0); "The letter O";  penpos1(thick,10); penpos2(.1[thin,thick],90-10);  penpos3(thick,180+10); penpos4(thin,270-10);  x1l=w-x3l=curve_sidebar; x2=x4=.5w;  y1=.49h; y2l=-o; y3=.51h; y4l=h+o;  penstroke z1e{down}..z2e{right}             ..z3e{up}..z4e{left}..cycle;  penlabels(1,2,3,4); endchar;   Now, when running the mf command and loading this file, I get a window that is unfortunately empty.  Supposedly a large letter "O" should appear.  $ mf This is METAFONT, Version 2.718281 (TeX Live 2013) **io (io.mf The letter O [79]) *end Output written on io.2602gf (1 character, 1724 bytes). Transcript written on io.log.  $ gftodvi io.2602gf  gftodvi: fatal: tfm file `gray.tfm' not found.   Then, when trying to use the gftodvi command I get the above error.   Any ideas why my "window" that should display the letter "O" is blank/empty? How do I resolve the gray.tfm error? Are the two related?   Note: I'm using Ubuntu 14.04, but the TeXLive installation is NOT from the Ubuntu repository.  I used the installer from  and created a local "portable" installation. It is often stated, particularly in popular physics articles and videos, that if one measures a particle A that is entangled with some other particle B, then this measurement will immediately affect the state of the entangled partner.  For example, if Alice and Bob share an entangled pair of electrons and Alice measures her spin in the $x$ direction, then Bob's spin will also end up spinning in that direction, and similarly if she measures in the $z$ direction. Moreover, the effect will be instantaneous, regardless of the spatial distance between the two particles, which seems at odds with special relativity.  Can I use a scheme like this to communicate faster than light? Can not find the tag library descriptor for “http://java.sun.com/jsp/jstl/core” For some reason I can't unhide my objects anymore in this file. Is it a bug? CPU on Ubuntu running high In the terminal I used "top" to check the processes and how much of the CPU was being used as I thought max was 100%, so I restored 292,000 photos with recoverjpeg on my sda1 HDD, I opened the folder and the Nautalis is  fluctuating between 130% and 170% , should I close the program and am I at risk of my gpu locking up ? Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” Reproduced from Exercise 2.2 of Nielsen &amp; Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):     Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\rangle$, and $A$ is a linear operator from $V$ to $V$ such that $A|0\rangle = |1\rangle$ and $A|1\rangle = |0\rangle$. Give a matrix representation for $A$, with respect to the input basis $|0\rangle, |1\rangle$, and the output basis $|0\rangle, |1\rangle$. Find input and output bases which give rise to a different matrix representation of $A$.   Note: This question is part of a series attempting to provide worked solutions to the exercises provided in the above book. How to change the date format? Conditional Expectaction 3 variables Suppose $X,Y$ and $Z$ are multivariate normal with means and full covariance matrix. The conditional expectation $E(X | Y)$ is well know. What is the conditional expectation of $E(X | Y,Z)$ if $Y$ and $Z$ (and $X$) are correlated? Standard textbooks only seem to cover the case when $Y$ and $Z$ are uncorrelated. Make clear which profile data is visible to whom Word style "keep with next" except on last item How to set up reverse tethering over USB? My HTC G2 phone is rooted and running Cyanogenmod 7 I don't have a data plan. Sometimes I want to connect the phone to the Internet when there isn't Wi-Fi, to update Market apps, backup SMS messages to Gmail, sync new contacts from Gmail, etc.  Things I can't do with USB mass storage mode. I have a Windows 7 Professional computer connected to the Internet, but I'm not allowed to set up an ad-hoc Wi-Fi network.  (If I do, they will notice and hunt me down.)   Is there a way for the phone to access the Internet through the USB connection to the computer?  If so, how do I set it up? Can a quasiclassical electron wave packet in elliptic orbit be formed from bound hydrogen-like eigenstates? Position probability densities of eigenstates of hydrogen-like systems have axial symmetry, so that the wavefunction too much resembles the circular orbits in Bohr's model. I'd like to have a demonstration of correspondence principle, where an electron would be localized to look somewhat like a classical particle, and move in an elliptic (non-circular) orbit around a nucleus.  It seems though that if we try to make a wave packet too localized, then it'll disintegrate (get scattered by the nucleus) too fast to see anything resembling an elliptic orbit. OTOH, if we take it too spread out, it'll have to be quite far from the nucleus so as to avoid hitting it, and thus it'll have high total energy, which may appear to be above ionization threshold (at least partially), after which it'd be quite hard to analytically calculate evolution of the wave packet.  Thus my question is: is it possible to form a more or less localized wave packet, which would (on average) move in an obviously-elliptic orbit (major/minor axes ratio of 4:3 or higher), and only require bound states to fully represent it? If yes, then what properties (FWHM, apocenter, angular momentum etc.) should it have for this to be possible? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved.