I'm trying to compute  $$ \lim_{x\to 0^+} x^{\sin(x)}. $$  I've been trying to get it into a form where I can apply L'Hopital's rule, but I haven't had any success. Namely, I haven't been able to massage it into a fraction form. Any advice?
I tried writing $y=x^{\sin x}$, so $\ln y=\sin x\ln x$. I tried to rewrite the function as $\ln(x)/\csc x$ and apply l'Hopital to the last function, but it's a mess.   Is there a by hand way to do it?
How can I enable "clicklock"? Suppose $f$ is entire and $|f(z)| \leq A + B |z|^{3/2}$. Show that $f$ is a linear polynomial.   My attempt: Since $f$ is entire, we know that it has a Taylor series expansion on the circle $|z| = R$, that is $f(z) = \sum_{n = 0}^\infty a_n z^n$. By using Cauchy's estimate, we can write   $|f^n(0)| \leq \frac{n! M}{R^n}$. Since this holds for any $R&gt;0$, it follows that for $n\geq 2$, $|f^n(0)| \leq 0$. This implies that $f^n(0) = 0$ for $n\geq 2$. Hence $f$ is a polynomial of degree at most $1$.  Edit 1 : On the circle $|z| = R$, $|f(z)|\leq A + B R^{3/2}$.  Thank you very much.  Kindly suggest me if I am wrong? Is there any other apporoach to solve such kind of problems? Hide text using css I have a tag in my html like this:  &lt;h1&gt;My Website Title Here&lt;/h1&gt;   Using css I want to replace the text with my actual logo. I've got the logo there no problem via resizing the tag and putting a background image in via css. However, I can't figure out how to get rid of the text. I've seen it done before basically by pushing the text off the screen. The problem is I can't remember where I saw it. If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$? Prove by induction $\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{n}\ge\frac{2}{3}n\sqrt{n}$ for all positive integers Assumption: $\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}\ge\frac{2}{3}k\sqrt{k}$   Prove true for $n=k+1$ $$\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}+\sqrt{k+1}\ge\frac{2}{3}(k+1)\sqrt{k+1}$$  I'm upto :$$\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}+\sqrt{k+1}\ge\frac{2}{3}k\sqrt{k}+\sqrt{k+1}$$after which I'm stuck. What is the correct name to use with /give for Lapis Lazuli? Preventing Brute Force Logins on Websites As a response to the recent  and , what is the best way to secure your website against brute force login attacks?  Jeff's post suggests putting in an increasing delay for each attempted login, and a suggestion in the comments is to add a captcha after the 2nd failed attempt.  Both these seem like good ideas, but how do you know what "attempt number" it is?  You can't rely on a session ID (because an attacker could change it each time) or an IP address (better, but vulnerable to botnets).  Simply logging it against the username could, using the delay method, lock out a legitimate user (or at least make the login process very slow for them).  Thoughts? Suggestions? I am solving the TOEIC workbook, by the way, I faced conflict that doesn't match meaning I knew.  The problem is     The newest branch of the library is located in/on Pine Street.   I think either 'in' or 'on' have the same meaning. But the commentary said that 'be located in' is mainly writing in front of building name and city name.   I don't know what is correct? Custom Gmail labels sort I want to sort the labels in my own order instead of the default abc order, so that, for example, the label named bills should come up first.  Any way to do that? What is the purpose of the 'nobody' user? After I read  I noticed that there is a user account named 'nobody' in my Ubuntu system.   Also I noticed that I can login in this account from terminal using the following command and my password:  sudo su nobody     It doesn't mind me at all, but I want to know what is the purpose of this user? Is it created by default on a fresh install of Ubuntu or is created by installing a particular package? Which day does "next Tuesday" refer to? At what point does next Tuesday mean      the next Tuesday that will come to pass     and no longer      the Tuesday after the Tuesday that will come to pass?   And, when does the meaning switch back? How to allow certain html tags? How to monitor process/program execution in windows? We are trying to develop a small application that can monitor the programs/processes that are executing in a windows machine.  If the program/process is not supposed to run, it should be blocked. It works similar to an antivirus.  This is the basic idea.  I want to know the ways to hook into the OS to get notified about every single program/process trying to run in the machine. HP PSC 1410 all scanner's images are black I have a HP PSC 1410 printer with scanner inside.  I've been scanning documents and images with it for the last couple of years.  Recently my scanner has stopped performing as excepted and i can't scan or copy images\documents.  My problem is that every images\document i'm trying to scan - the result from the printer is a black page.  I've been googling about this issue and couldn't find any helpful stuff beside replacing the whole printer - which is something i'm trying to avoid.  Any ideas ? I've read something about the shipping lock of the scanner but couldn't find any information about how to check it.. What is it called when a non-verb is used as a verb? This is in very common use on the internet. I just read this:     "[The] hotel employee walks in, I say "uh, puppy" and she just NOPEd the [heck] out of the room."   This is often done by adding 'd to the word. Get last n lines of a file, similar to tail I'm writing a log file viewer for a web application and for that I want to paginate through the lines of the log file.  The items in the file are line based with the newest item at the bottom. So I need a tail() method that can read n lines from the bottom and support an offset.  This is hat I came up with: def tail(f, n, offset=0):     &quot;&quot;&quot;Reads a n lines from f with an offset of offset lines.&quot;&quot;&quot;     avg_line_length = 74     to_read = n + offset     while 1:         try:             f.seek(-(avg_line_length * to_read), 2)         except IOError:             # woops.  apparently file is smaller than what we want             # to step back, go to the beginning instead             f.seek(0)         pos = f.tell()         lines = f.read().splitlines()         if len(lines) &gt;= to_read or pos == 0:             return lines[-to_read:offset and -offset or None]         avg_line_length *= 1.3  Is this a reasonable approach?  What is the recommended way to tail log files with offsets? Can a virtualized machine have the CPU and RAM resources of multiple underlying physical machines? We are studying implementing some virtualized servers here, but we don't know what will be better suitable for us. Some folks are saying better have two huge servers, and others are saying have like a ten middle-end servers.  We have a legacy Visual Foxpro application, which nowadays run on Dual Xeon E5405 @ 2GHz and 16Gb of RAM. The currently server its getting too slow due the number of active users and process running on it. Virtualizing this server will give us the benefit of an faster disaster recovery.  So the question is, having like ten physical servers running at 1.7GHz and 4Gb of RAM, we could virtualize one server into 4 machines, and have one virtualized server running at 6.8GHz and 16Gb of memory? If yes, there is some how if one machine stops, automatically manage this virtual machine to another one, and execute the appropriate maintenance on it, and later back to it again? Proving that if $G$ has a Hamiltonian circuit, then the line graph $L(G)$ also has a Hamiltonian Circuit I have a graph G = (V, E). I know that the line graph L(G) of G is defined as follows: each vertex in L(G) corresponds to an edge in G, and two vertices are connected by an edge in L(G) if and only if the corresponding edges in G are adjacent.  Now I am trying to (a) prove that if G has a Hamiltonian circuit, then L(G) also has a Hamiltonian circuit, and (b) give an example of a graph G such that G does not have a Hamiltonian circuit but L(G) does.  However, I'm extremely stuck on how to do this, and any help would be greatly appreciated. Thanks so much! What is the fastest way to convert float to int on x86 What is the fastest way you know to convert a floating-point number to an int on an x86 CPU.  Preferrably in C or assembly (that can be in-lined in C) for any combination of the following:   32/64/80-bit float -> 32/64-bit integer   I'm looking for some technique that is faster than to just let the compiler do it. I am doing some arcpy -functions and afterwards I'm trying to add the calculated results from one table to another table. I'm using the arcpy.AddJoin_management() function.   As input table I'm using a table which is stored in a personal geodatabase. The personal geodatabase is not the workspace environment! As "JoinTable" a Table of a file geodatabase is used. This file geodatabase is the workspace geodatabase environment.  The initialized function looks like this:  (%self.getWorkspaceEnvironment => "W:/a18fb/data/mypersonalgeodatabase.mdb" and tFactsheetHochwasser => table in the personal geodatabase)  arcpy.AddJoin_management("%s/tFactsheetHochwasser"%self.getWorkspaceEnvironment(),"KPG", "GebaeudeKtn_BWVred1Stat","KPG","KEEP_ALL")  If the code is executed the following error-message appears:  Failed to execute.Parameters are not valid. The value cannot be a table     ERROR 000840: The value is not a Raster Layer.      ERROR 000840: The value is not a Raster Catalog Layer.      ERROR 000840: The value is not a Mosaic Layer.      Failed to execute (AddJoin)`.