Hint on finding limit without application of L'Hospital's rule (mathematical analysis)
Limit $\lim\limits_{x\to0}{\frac{\ln{(e+x)}-e^x}{\cos^2{x} -e^x}}$
I live in a town where fresh or whole milk is not readily available. I was wondering if I could make cheese from UHT Milk. If so, what kind of cheese and what method I should use? Or rather, Should I? I've noticed that if I downvote an answer, the record of the downvote is visible to me in my profile.  But when I view my profile from another browser that's not logged into Stack Overflow, I've noticed that I don't see the record of the downvote.  So I was wondering if the downvote record is visible only to me?  If so, is this done to protect downvoters from retaliation? "have" vs."have got" in American and British English I have looked through several questions and answers on EL&amp;U, and often there is an indication that American English prefers "have" while British English prefers "have got". In addition, there are several references to "have got" being more informal than "have" (e.g.: , ). But where is it considered more informal? In American English alone or both American and British English?   This point may seem obvious to those who made the above mentioned references, but the fact is that even the most fluent users of English in my country will say without a doubt that 'have got' is the preferred usage in the UK and, therefore, it must also be more formal. Why? Because in an academic environment, the more informal structures are generally frowned upon, outside some specific spoken exercises. And if the teachers insist on using "have got", it follows it isn't informal.  So, is the teaching of (British) English in our schools transmitting the wrong idea?  EDIT: I'm adding some information in answer to the comments.  I'm mainly concerned with the idea of possession:      I have got a cat vs. I have a cat and She has got a dog vs. She has a dog   As mentioned in the comments, every (Portuguese) student will be told that "have" and "have got", when it comes to the idea of possession, are absolute synonyms, the only two differences being:    "have got" is preferred by British English, "have" is preferred by American English   the negative and interrogative structures differ    I have never heard of any distinction in formality. In fact, I was rather surprised when I went through the EL&amp;U archives and read about it.  So, would "I have got a cat" be as formal/informal as "I have a cat"?   And how do British and American English look at its formality/informality? Is it the same? What is the Necessity of $...$ Around Numbers? I typed up an assignment with a lot of numbers in LaTeX and turned it in to my adviser.  When he returned it to me with some corrections, one of which was a lot of $ around my numbers.  I understand that $ are necessary around some mathematical formulas, but I was just using numbers like -2 and 4.  I did not see any change in the font of the numbers when I added the $ either.  Since my adviser did not want to give me an explanation, what is the rationale? Do electrons have shape? Convert binary to ASCII and vice versa Proof of Frullani's theorem How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are  Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0, $$ and let $ a,b \in \left( {0,\infty } \right)$. Prove that $$ \int\limits_0^{\infty}  {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}} {x}}dx  = f\left( 0 \right)\left[ {\ln \frac{b} {a}} \right] $$  If you know a more general version please give it to me )= I can´t prove it. Leftmargin problem at the bibliography I have some trouble with the bibliography alignment. It's similar to , but that solution doesn't work for me.   Here is MWE:  \documentclass{article}  \let\oldthebibliography=\thebibliography \let\endoldthebibliography=\endthebibliography \renewenvironment{thebibliography}[1]{%   \begin{oldthebibliography}{#1}%   \setlength{\parskip}{0ex}%   \setlength{\itemsep}{0ex}%   \setlength{\itemindent}{3ex}   \setlength{\leftmargin}{0pt}% }% { \end{oldthebibliography}% }  \begin{document}  Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.    Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.   \begin{thebibliography}{0} \bibitem{1} Baider A. Noncompact Riemannian manifolds with discrete spectra // J.Diff.Geom. -- 1979 -- V. 14 -- p. 41--57. \bibitem{2} Berger M., Gauduchon P., Mazet E. Le spectre d'une vari\'et\'e Rieman\-nienne -- Berlin-New York: Springer-Verlag, 1971. -- 251p. -- (Lecture Notes in Math.; V.~194) \end{thebibliography} \end{document}   So I need the reference labels to be aligned with the parindent, and the rest part of the each bibliography entry should have general text width. I thought it's enough to set \leftmargin=0pt, but it didn't work. I have the following result:   And what is actually more weird - it seems any value of \leftmargin doesn't have any effect to the bibliography display. The solution from the link above works fine if we don't have biblabel, but it's not my case.  Could you help me to move lines with the red marks to the green one? I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space? Heisenberg Uncertainty Principle scientific proof states that:    $$\sigma(x)\sigma(  p_x )\ge \frac {\hbar}{2}.$$  What is the scientific proof of this principle? C array setting of array element value beyond size of array Let G be a finitely generated abelian group and M a compact manifold,  I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $.  First I was thinking if  I could do induction over $r$ because for $r=0$, $H_0(M,G) \cong \bigoplus_{i=1}^nG$ where $n$ denotes the number of connected components of M.But for $r&gt;0$ I can't see a way to relate $H_r(M,G$) and $H_{r+1}(M,G)$.  Then I tried to use the fact that for every compact surface there exists a finite cover $\{A_i\}_{i=1}^m$ such that the finite intersections of $A_i$'s are contractile.Let $\{A_i\}_{i=1}^m$ such cover for $M$.Then,using induction on $m$, we have :  -For $m=0$, $M$ is contractile and $G$ finitely generated then $H_r(M,G) $is finitely generated.  -For $m=r$ it's true by induction hypothesis.  -For $m=r+1$ now putting $A_{r+1}=B$ and $\cup_{i=1}^r A_i=A$, and using Mayer Vietoris we get:  $$ ...\rightarrow H_{r+1}(A\cap B,G)\rightarrow  H_{r+1}(A,G) \oplus H_{r+1}(B,G) \rightarrow H_{r+1}(M,G)\rightarrow H_{r}(A\cap B,G) \rightarrow...$$   Here I don't know what to do, maybe I'm missing some algebraic argument It's easy to see that $H_{r+1}(A\cap B,G)$ and $H_{r+1}(A,G) \oplus H_{r+1}(B,G)$ are finitely generated.But how can I argue that $H_{r+1}(M,G)$ is finitely generated? Any idea would be aprecciated. Calculating grid convergence (True North to Grid North) I would like to know how to calculate the grid convergence angle between True North and Grid North for UTM maps. I am using ArcGIS 10.0 (ArcView licence). and have used the  tool from the Cartography toolbox. However, I would like to know how to calculate this angle myself (using Excel) to compare against the ArcGIS result? I'm fairly certain this has happened to all of us at some point: You go into a room to get something, but once you're there, you can't remember what you intended to get. It seems like a specific, common phenomenon like this must have a name, but I can't think of it. Does anyone know a name (either formal or slang) for this?  I don't want a general term like forgetfulness or absent-mindedness, which could apply to all sorts of forgetting, or a term for not being able to remember a certain word. I'm looking for a word or phrase that describes this one specific situation of forgetting what you need to get as soon as you go to get it. This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? How to track the number of tag wikis I edited? I want to see the number of tag wiki that I edited till now. Also I want to see which are accepted and which are rejected with the detail of all those tags. How can I know that? Also I want to know the progress for Research Assisstant badge. How do I create a PDO parameterized query with a LIKE statement? Here's my attempt at it:  $query = $database-&gt;prepare('SELECT * FROM table WHERE column LIKE "?%"');  $query-&gt;execute(array('value'));  while ($results = $query-&gt;fetch())  {     echo $results['column']; } How to sum up values of each two rows across their line in linux Show that $a^2 \bmod b = (a \bmod b)^2 \bmod b$ Show that  $$a^2 \mathrm{\ mod \ } b = (a \mathrm{\ mod \ } b)^2 \mathrm{\ mod\ } b$$ for $ a, b \in \mathbb{Z}^+ $.  this was derived from an Informatics olympiad question. Reputation widget crash I have an issue with crashing reputation widget when it's added to the home screen. When I add it for the first time, it says "Please login before using this widget" (even though the app itself is logged in already). So I open the app, go back to the home screen, add the widget again and it crashes (and it's not added to the home screen). When I'll try to add it again, it says "Please login before using this widget" again. And then it just repeats - "Please login...", execute the app, crash, "Please login...", etc.  Restart of the device didn't make any change.  I have found .  My set up:  Stack Exchange app version: 1.0.89 Android version: 5.0.1 Device: Samsung Galaxy S4