Calculate limit of function 4 Why  \begin{equation} \lim_{x\to-\infty}\sqrt{x^2-x-1}-x=+\infty, \end{equation} Thanks
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$ This is being asked in an effort to cut down on duplicates, see here: , and here: .    What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$  In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} +\cdots +a_1 x+ a_0$, how would I find $$\lim_{x\rightarrow\infty} P(x)^{1/n}-x.$$  For example, how would I evaluate limits such as $$\lim_{x\rightarrow\infty} \sqrt{x^2 +x+1}-x$$ or $$\lim_{x\rightarrow\infty} \sqrt[5]{x^5 +x^3 +99x+101}-x.$$
I'm positive that vs I strongly believe that In general, which one is conveys a stronger conviction/opinion? I did some quick google search but didn't find any interesting results.      I'm positive that   or     I strongly believe that How can I use GDAL to batch define a projection? What are the various units (ex, em, in, pt, bp, dd, pc) expressed in mm? How can one determine the length of the different units (measured 1ex, 1em, 1in, 1pt, 1bp, 1dd, 1pc) in mm? I need to analyse data and am unsure whether or not I should use a multinomial logistic regression analysis or a binary logistic regression analysis. I have just conducted a latent class analysis and am interested in seeing if my variables can predict class membership. There are only two classes and so my response variable only has two categories; however the variables range from having 2 to 3 to 4 and are mostly nominal categorical data. It would be easier to ask for a user's password only once during registration.  The problem: The user could make a mistake while typing the password once because of hiding letters.  The solution: The user could have a toggle button for showing or hiding the password.    Working example with . This approach could be used on the registration or login page.   Are there any benefits to asking a user's password twice during registration vs just not masking the password? Why would you ask twice?  P.S. :        Users make more errors when they can't see what they're typing while filling in a form. They therefore feel less confident. This double degradation of the user experience means that people are more likely to give up and never log in to your site at all, leading to lost business. (Or, in the case of intranets, increased support calls.)   The more uncertain users feel about typing passwords, the more likely they are to (a) employ overly simple passwords and/or (b) copy-paste passwords from a file on their computer. Both behaviors lead to a true loss of security.      Update: I created a . So you may use it if you want to.    Update 2: .  Update 3: Internet Explorer 10 added a toggle password visibility icon. It looks like this:    Update 4: Article about .  Update 5: Example with . How do I find out which version and derivative of Ubuntu is right for my hardware in terms of minimal system requirements? How can I swap Ctrl and Fn keys on a corporate computer? Our computers have been upgraded to HP EliteBook 8570p machines and while they're very nice machines, they are also very large -- so large that the left Ctrl key is too far away, making Ctrl+V really difficult to do with one hand.  I'm sure there are ways to re-map the keys so that their signal is swapped. But add the factor of corporate security, is there still a reasonably simple approach to this?  I'm sure you'd like more information before you can answer -- I'll try to add updates based on comments and answers. Cannot use polyglossia/babel with biblatex in LuaTeX: etoolbox Toggle undefined This is an example test.tex document. Both babel and polyglossia failed and produce the same error: \documentclass{book}  %\usepackage[vietnamese]{babel} \usepackage{polyglossia} \setdefaultlanguage{vietnamese}  \usepackage{biblatex}   \title{Sth}  \begin{document}  \maketitle  \end{document}   is the full log. And my config for VS Code: &quot;latex-workshop.latex.recipes&quot;: [     {         &quot;name&quot;: &quot;lualatex-&gt;biber&quot;,         &quot;tools&quot;: [             &quot;lualatex&quot;,             // &quot;biber&quot;,         ]     } ], &quot;latex-workshop.latex.tools&quot;: [     {         &quot;name&quot;: &quot;lualatex&quot;,         &quot;command&quot;: &quot;lualatex&quot;,         &quot;args&quot;: [             &quot;--shell-escape&quot;,             &quot;-synctex=1&quot;,             &quot;-interaction=nonstopmode&quot;,             &quot;-file-line-error&quot;,             &quot;-pdf&quot;,             &quot;%DOC%&quot;         ]     },     {         &quot;name&quot;: &quot;biber&quot;,         &quot;command&quot;: &quot;biber&quot;,         &quot;args&quot;: [             &quot;%DOCFILE%&quot;         ]     } ],  I installed all Vietnamese-related package I found using tlmgr search, notably babel-vietnamese and vntex, to no avail. When I use English language, it works. When I paste the exact document to Overleaf, it also works. So it's something on my part. How can I fix this? Debate, Bare Plurals, and Mass Nouns When is $(p - 1)! + 1$ a power of $p$? A friend asked me this question:     If $p$ is a prime, prove that $(p - 1)! + 1$ is a power of $p$ if and only if $p = 2, 3$ or $5$.   Clearly one direction is obvious, namely that $p=2,3,5$ implies $(p - 1)! + 1$ is a power of $p$.  The other direction is not clear to me. Since by Wilson's theorem $p$ divides $(p - 1)! + 1$ so we need to show that if there are no other prime factors then $p=2,3,5$. Can someone give me a hint for establishing this?  Thanks Why does the order in which libraries are linked sometimes cause errors in GCC? How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! Prove the following equality using mathematical induction: Prove the following equality using mathematical induction: For any integer $n \ge 1$  $$\sum_{i=1}^{n} \frac{1}{i(i+1)} = \frac{n}{n+1}$$  I understand for the base base I need to have $n=1$. If I substituted $n$ for 1 and $i$ for 1 I would get   \begin{align} \sum_{i=1}^{1} \frac{1}{1(1+1)} &amp;= \frac{1}{1+1} \\ \sum_{i=1}^{1} \frac{1}{2} &amp;= \frac{1}{2} \end{align}  As for the inductive step, I inputted $n +1$ but it did not work for me because honestly I did not know what to do with the $i$ and what I should do next.  How would I go about doing the next step and an explanation for $i$ would be helpful. Facebook changed how their API serves profile pictures that are requested with a non-application scope UID. Now because before Graph API 2 all applications shared the original UID, if users like me has joined Stack Exchange websites before  2, rolling out their profile pictures are not shown properly. Facebook suggests to use an access token to query those profile pictures, either an application access token or a user access token. A user access token is not a reliable option at all, because it would only work for users who joined website with Facebook and are currently logged in. Application access tokens, by the way, can be used by creating sort of a proxy that would retrieve a profile picture from Facebook and display it. Yet still it's not perfect. I think the best solution is to communicate with Facebook and ask them to offer an opportunity to convert old Graph API 1 UID's to application scope user IDs. Why won't my villagers breed How do I restrict my kids' computing time? Replace $ with £ using cat Command I have file test_output_archive.svd with some Dollar signs ($). I have generated a shell script Replace_dollar_pound.sh with cat command to replace dollar symbol with Pound (£) script is working on Solaris 10, Dollar getting replaced as expected. In Solaris 11, but the dollar is not getting replaced, not throwing any errors  Content of test_output_archive.svd : Line Rental $9  OOB Only for $10  Replace Dollar symbol with Pound at step1, Remove junk charactes at step 2, Remove temp file at step3, Rename at step4**  Content of Replace_dollar_pound.sh : cat test_output_archive.svd | tr "\044" "\243" > temp_archive.svd cat temp_archive.svd | tr -d "\302" > test_output_archive2.svd rm temp_archive.svd mv test_output_archive2.svd test_output_archive.svd  Any Modifications to be done as OS is updated to Solaris 11 ?? Plain integer partitions of $n$ using $r$ parts Division of number $n$ on parts $a_1,...,a_r$ where $a_1 \le ... \le a_r$ we call a plain if $a_1 = 1$ and $a_i - a_{i-1} \le 1$ for $2 \le i \le r$. Find enumerator (generating function) for plain divisions.  my try  The hint was to use bijection between plain divisions and some commonly know enumerator. I tried to use enumerator of divisions on different parts: $$ (1+x)(1+x^2)...(1+x^r)$$ where number of plain divisions is  $$[x^n](1+x)(1+x^2)...(1+x^r) $$ let function  $$f(n,r) = [x^n](1+x)(1+x^2)...(1+x^r) $$  For some first divisions it works. For example: $$f(4,3) = 1 $$ $$f(6,3) = 1 $$ $$f(11,5) = 2$$ But when I tried to find bijection, I failed. I found that this function isn't correct because $f(15,6) = 4$ but should be equal to $3$ because: $$15 = 1,1, 2, 3, 4, 4 \\ 15 = 1, 2, 2, 3, 3, 4\\ 15 = 1, 2, 3, 3, 3, 3 $$. There I stucked. I use 'Helvetica' as my font and build my document using Texmaker. But the PDF generated has jagged edges while using this font and when I use a different font (say Times or even Helvetica sans-serif) it builds just fine.     Please suggest if there is anyway I can get over this. The problem with these jagged edges is that it appears very dull and unreadable at 100% resolution.  OK guys, now here is the MWE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% HEADER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[a4paper,twoside,12pt]{book}  %% Language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} %\usepackage[utf8]{inputenc} \usepackage{graphicx,color,subfigure}  %% Line Spacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{setspace} %\singlespacing        %% 1-spacing (default) %\onehalfspacing       %% 1,5-spacing \doublespacing        %% 2-spacing  %% Other Packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[left=1.5in, right=1.5in, top=1in, bottom=1in, includefoot, headheight=13.6pt]{geometry} \usepackage[pagebackref=true,breaklinks=true,letterpaper=true,colorlinks,bookmarks=false]{hyperref} \usepackage{helvet} \usepackage[square,numbers]{natbib} \usepackage{hyphenat} \usepackage{fancyhdr}   \begin{document}  \pagestyle{empty} %No headings for the first pages.  \cleardoublepage %The first chapter should start on an odd page.  \pagestyle{plain} %Now display headings: headings / fancy / ...  \pagebreak  %------------------------------------------------------------------------ \chapter{Divorce}   A major change that has occurred in the Western family is an increased incidence in divorce. Whereas in the past, divorce was a relatively rare occurrence, in recent times it has become quite commonplace. This change is borne out clearly in census figures. For example thirty years ago in Australia, only one marriage in ten ended in divorce; nowadays the figure is more than one in three (Australian Bureau of Statistics, 1996: p.45). A consequence of this change has been a substantial increase in the number of single parent families and the attendant problems that this brings (Kilmartin, 1997).   \end{document} How did Rutherford conclude that most of the mass (as well as the positive charge) was concentrated in the nucleus? Geiger and Marsden's experiment led Rutherford to believe that the positive charge and most of the mass of the atom was concentrated in a small region. I understand what led him to conclude the way the positive charge is positioned in the atom. But how did he conclude that most of the mass was in a small region (the nucleus)?  How did the distribution of the mass matter after all? Given that the electric force is greater than the gravitational force by many magnitudes, the force between the positice charge and the electrons was predominantly electric.   So how did Rutherford conclude that most of the mass is in the nucleus?