"Two types of user" or "Two types of users"?
Types of things vs. types of thing
My son is 5 years old.   Over the last few months I have been observing some disturbing sexual behavior from my child.  He is rubbing his penis on  the bed or pillow. Sometimes he would get naked and rub very tightly and make some sounds of pleasure! Previously, I thought he was playing... but later I understood.   His father doesn't want me to scold him, but one day I was really surprised and shocked by one incident... when I was in deep sleep in the afternoon, I suddenly woke up and saw he had pulled up my skirt and trying to mount me! I was really shocked!  I'm really upset.  What should I do? Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$ Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm. If the distribution of test statistic is bimodal, does p-value mean anything? These questions are really bothering me. Any help would be much appreciated.  Let $p$ be prime  a) If $(G, \oplus)$ has order $2p$, prove that every proper subgroup of $(G, \oplus)$ is cyclic  b) If $(G, \oplus)$ has order $p^2$, prove that $(G, \oplus)$ has a subgroup of order $p$ Storing the original document class page layout with geometry Is it possible, using the geometry package, to store (and restore) the original layout specified by the document class (in my case, the class is report)?    Here is some context.  I have been editing a long document in the default page layout for the report document class.  Now I found out that I need a title page.  There is a particular title page layout, and in fact, it works perfectly if I just load the geometry package with its default and change just two settings.  But then I want the rest of the document back in the original page layout!  I can't do something like \restoregeometry, since I think geometry just overrides the class options... i.e. there doesn't seem to be anything to restore.  Similarly, it seems I can't use \savegeometry or \loadgeometry.  So how can I get back to my original layout for the rest of the document without painstakingly hard-coding in all of the specific page layout variables?    Here is a minimum working example:   \documentclass[a4paper,10pt]{report}  \RequirePackage[a4paper,dvips]{geometry} % g1 \geometry{top=2.5cm,bottom=2.3cm}        % g2  \begin{document}  \input{the_title_page}  \SomeGeometryCommandsHere  % g3  \input{the_rest}  \end{document}   If I run the above code with the lines labeled "g" commented out, then the title page does not have the correct page layout, but the rest of the document is formatted perfectly.    If I leave lines g1 and g2 uncommented, then the title page has exactly the correct layout, but no matter what I put for g3 (e.g. \loadgeometry, \restoregeometry, etc.) I can't recover the original layout that I had when g1 and g2 were commented.  Of course, I could completely specify the original layout exhaustively in line g3, and this should in principle work.  However, I spent almost one hour on this last night and when I still couldn't get it formated correctly I decided I really needed a smarter solution. FireFox 66 Quantum disabled all my extensions at midnight! This morning I woke up to:    and the FireFox help page doesn't really help...  Help!  What should I do now??? I need NoScript and AdBlock Plus to improve my surfing speed!!! 0:-) $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.  Just making sure I am on the right track...  Do I expand each piece out and solve algebraically? Vector bundle on a quadric $Q$ Real world use cases of bitwise operators Best Place to put Feature Requests? The only place I am aware of is the . If you have two envelopes, and ... "High response time" warnings in Google Webmaster Tools In my Google Webmaster Tools account, there are warnings about my site.     Some URLs listed in this sitemap have a high response time. This may indicate a problem with your server or with the content of the page.   What is the solution? Really stuck with this, \begin{cases}  2tx'(t)-1=x^2t \\  x(t=1)=1 \\  \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :( Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? The indexer into Dictionary throws an exception if the key is missing.  Is there an implementation of IDictionary that instead will return default(T)? I know about the TryGetValue() method, but that's impossible to use with LINQ. Would this efficiently do what I need?: myDict.FirstOrDefault(a =&gt; a.Key == someKeyKalue);  I don't think it will as I think it will iterate the keys instead of using a Hash lookup. Centering in the \align* environment What is the origin of Dol Goldur, and why did Sauron start there? Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? Why was the snaphat not awarded? Or why was it awarded incorrectly? There have been a few cases in which the  was not awarded.   There have also been other cases in which it was awarded incorrectly, on old questions.   What gives? Is it possible to remove a terminal command? For example, if there was a command call lookup (not an alias), how would I go about removing it? Exam with 18pt main font