Proof that $\lim_{x \to 0} \frac{\sin(x) - x}{x^2} = 0$ (Without L'Hospital)
$\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor
Changing default \citet font in biblatex I'm using biblatex to manage my citation in a natbib style :  \usepackage[style=alphabetic,maxnames=6,natbib=true]{biblatex}   However when I use the \citet command, the author's name is in a \textsc font, I would like to change this behavior so that the font used is the default one, but I could not find how to do this in biblatex's documentation. Does registering [name]overflow.com violate Stack Overflow's trademark? How do I install Java? How do I install Java, including in my browser, so websites with Java work? The new top bar layout is fine, but the black color is very visually jarring/intrusive.  Is using a lighter color an available option? I am trying to do kriging by gstat package through R.   Here is my Code:  krg&lt;-krige(var ~ 1,data,grid,model=vgm,nmax=30)   But each time I recive this warning:  1: In predict.gstat(g, newdata = newdata, block = block,  ... :    Warning: Covariance matrix (nearly) singular at location [1,4501,0]: skipping...   and when I try to plot it by spplot this error appears:  Error in seq.default(zrng[1], zrng[2], length.out = cuts + 2) :    'from' cannot be NA, NaN or infinite In addition: Warning messages: 1: In asMethod(object) :   complete map seems to be NA's -- no selection was made 2: In min(x) : no non-missing arguments to min; returning Inf 3: In max(x) : no non-missing arguments to max; returning -Inf   Here is the full code: You can download my data through this link:  and here is my full code:  library(gstat)                                          library(sp) library(raster)  coordinates(Mydata) &lt;- ~East + North proj4string(Mydata) &lt;- CRS("+proj=utm +zone=10+datum=WGS84") ext &lt;- as(extent(Mydata), "SpatialPolygons") r &lt;- rasterToPoints(raster(ext), spatial = TRUE) proj4string(r) &lt;- proj4string(Mydata)  v1 &lt;- variogram(cpue_female ~ 1, Mydata) vgm_mod_omni &lt;- vgm(psill = 0.0003, model = "Mat", range = 50000, nugget = 33, kappa = 0.5) G1 &lt;- krige(cpue_female ~ 1, Mydata, r, vgm_mod_omni, nmax = 30) gridded(G1) &lt;- TRUE spplot(G1) How to modify existing, unpushed commit messages? How to typeset subscript in usual text mode? It's easy to make subscripts in math mode: $a_i$.   How do I make a subscript outside math environment, likethis? What's the purpose of two capacitors in parallel? What's the purpose of the two capacitors in parallel on each side of the regulator in      I've seen similar setups in other similar circuits and can guess that it's related to one being polarized an one not, but I don't really understand what's going on there. How to deal with travel excitement? How does one usually deal with excitement about travel, prior to travel ? A lot of people feel them butterflies in their stomach before a big travel is coming up. Is there an effective way to put these feelings away ? Or how can one best deal with such excitement/feelings (which almost cause interference with day to day activities) ? \documentclass{article} \usepackage{algorithm}  \begin{document} \setcounter{section}{54} \section{A section with no label} \begin{algorithm} \label{myalg} \caption{An algorithm with the label `myalg'.} \end{algorithm} Reference to myalg: \ref{myalg}. \end{document}   With this code, the reference givens 55 (the section number); if I move the label after the caption command, I get the correct reference (1).  Why should this be the case? I find it confusing: when thinking about an algorithm as an environment, you expect to be able to label it right after you began it. At the very least I would expect some kind of error or "??" to appear, rather than the previous environment or reference-able entity being used. How to do joins in LINQ on multiple fields in single join CNAME for top of domain? Is it possible to set a CNAME record at the top of a domain? (i.e. @ CNAME www, @ CNAME foobar.com., etc.)  My ISP says that it's only possible to use CNAME's for subdomains but I've read somewhere else that is should be possible even if not recommended. Is there any way I can specify a standard or custom numeric format string to always output the sign, be it +ve or -ve (although what it should do for zero, I'm not sure!) Find the Values of $p$ and $q$ such that $\int_0^{\infty} x^p\ln(1+x)^q dx$ Converges or Diverges. The question is to find the value values of $p$ and $q$ such that the improper integral converges or diverges.  My friend indeed found some values of $p$ and $q$ such that it converges, but after many trials, I am still getting divergent for all values.  Via L'Hospital, we have the following:  $$\lim_{x \to 0} x^c\ln(1+x)^q = \infty \space \text{when} \space  c &lt; 0$$ $$\lim_{x \to 0} x^c\ln(1+x)^q = 0 \space \text{when} \space  c &gt; 0$$ $$\lim_{x \to \infty} x^c\ln(1+x)^q = \infty \space \text{when} \space  c &gt; 0$$ $$\lim_{x \to \infty} x^c\ln(1+x)^q = 0 \space \text{when} \space  c &lt; 0$$  Thus, $$\int_1^{\infty} x^p\ln(1+x)^q dx = \int_1^{\infty} x^{p-c}x^c\ln(1+x)^q dx &gt; \int_1^{\infty} x^{p-c} dx$$ which diverges when $p &gt; -1$.  Here, we set $c = \frac{1}{2}(p+1) &gt; 0$.  On the other hand, $$\int_0^1 x^p\ln(1+x)^q dx = \int_0^1 x^{p-c}x^c\ln(1+x)^q dx &gt; \int_0^1 x^{p-c} dx$$ which diverges when $p &lt; -1$.  Here, we set $c = \frac{1}{2}(p+1) &lt; 0$.  Thus, the integral diverges whenever $p \ne –1$.  Then, let $p = -1$.  $$\int_0^{\infty} \frac{1}{x}\ln(1+x)^q dx = \int_0^1 \frac{1}{x}\ln(1+x)^q dx + \int_1^{\infty} \frac{1}{x}\ln(1+x)^q dx $$ $$ \ge \int_0^1 \frac{1}{x+1}\ln(1+x)^q dx + \int_1^{\infty} \frac{1}{x+1}\ln(1+x)^q dx $$  The first summand diverges when $q &lt; -1$ and the second diverges when $q &gt; -1$.  Finally, let $ p = -1$ and $q = -1$.  The integral $\int_0^{\infty} \frac{1}{x\ln(1+x)} dx ≥ \int_0^{\infty} \frac{1}{(x+1)\ln(1+x)} dx$ diverges on both ends.  Did I make a mistake? On proving that a finitely generated group has a finite number of subgroups with index $n$. Theorem: If $G$ is a finitely generated group, then it has a finite (maybe zero) number of subgroups of index $n$ for any $n\in \mathbb{N}$.   is a sketch of a proof. It consists of assuming such a subgroup $H$ exists, and allowing $G$ to act on $G/H$ by left multiplication. Such action defines a homomorphism $$\phi :G \rightarrow \text{Sym}(G/H)\cong \mathbb{S}_n$$  If $G=\langle g_1, \ldots ,g_r\rangle$, then the homomorphism is uniquely determined by where it sends the generators $g_i$. In particular, there are at most $(n!)^r$ such homomorphisms.  The last step of the proof requires one to notice that a different subgroup $H'$ of index $n$ in $G$ would produce a different homomorphism, and here is where I'm struggling. I've been able to show that $\text{ker}(\phi )=\text{core}(H)$, yet haven't been able to rule out the possibility of $\text{core}(H)=\text{core}(H')$. Moment of inertia: why $\mathrm{dI}=r^2\mathrm{dm}$ instead of $\mathrm{dI}=m\mathrm{dr^2}$? Rather than rephrasing my question, let me describe to you the desired user-case:  I create a short shell-script to run command "gnome-terminal --someoptionflagname 'my text to be posted'", and execute this script.  Gnome-terminal pops up, with command-line prompt followed by my text.   ie: fields@mycomputer:/$ my text to be posted  Can this be done? What is the Big-O of a nested loop, where number of iterations in the inner loop is determined by the current iteration of the outer loop? What is the Big-O time complexity of the following nested loops: for (int i = 0; i &lt; N; i++) {     for (int j = i + 1; j &lt; N; j++) {         System.out.println(&quot;i = &quot; + i + &quot; j = &quot; + j);     } }  Would it be O(N^2) still? Inequality $(1+\frac1k)^k \leq 3$ How can the following inequation be proven?  $$a^2 + b^2 + c^2 \ge ab + bc + ca$$