Is $\frac{\sin(x)}{x}$ continuous at $x=0$? Whats the value at $x=0$? Is $\dfrac{\sin(x)}{x}$ at $x = 0$  continuous?  Whats the value at $x=0~?$
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it... Which are the word orders that can be found in English? Besides , which are the  that can be found in English? Are there any that are peculiar to dialects such as Singlish or Indian English? Please provide an example sentence for each order along with pertinent information such as tense, mood, voice, etc. Why can't polynomials have negative exponents or division by a variable Why can't:  $$2x^{-3} - 3x$$  or  $$\frac{1}{2x}$$  be polynomials too?  Why have a definition that excludes these algebraic forms? How to delete specific lines from a txt file I have a Bash script that writes data into a log file, then from that I take out stuff I want using the sed command. That gives me a file data.txt. What I want to know is how I go about deleting specific line inside that data.txt file.  For example:  123 456 789   I want to remove the 2nd line, containing 456 so that I only have  123  789   I tried with sed '2d' data.txt but it doesn't work.  I use this command to create data.txt from log.log:  sed -nE '1s/.{1}(.{2}).*/\1/p;' log.log &gt;&gt; data.txt Unique (non-repeating) random numbers in O(1)? I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible? How to delay a program such as cairo-dock (No openGL) or Firefox?  Sometimes programs crash when started with other startup apps. I want programs to start 10 seconds late to avoid a crash. Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse? Problem 3. Show that for a sequence $(x_n)$ the following are true:   (i) $\lim x_n=0$ if and only if $\lim |x_n|=0$.   (ii) $\lim x_n=L$ implies $\lim |x_n|=|L|$. Is the converse true? Prove or give a counterexample.    (i) is already done, easy.  I'm halfway done with (ii), I split it into three cases:  $L = 0$, $L &gt; 0$ and $L &lt; 0$.  For $L = 0$ I just refer to part (i).  For $L &gt; 0$, we know that $|x_n - L| &lt; \epsilon$   if lim $|x_n|$ = $|L|$, then we must have  $||x_n| -|L|| &lt; \epsilon$  but by the reverse triangle equality, $||x_n| - |L|| &lt; |x_n - L|$ so clearly $||x_n|-|L|| &lt; \epsilon$, thus  lim $x_n = L$ $\implies$ lim $|x_n| = |L|$ for $L &gt; 0$  for some reason I'm confused as to part three, $L&lt;0$. I'm having trouble seeing the difference between $L&gt;0$ and $L&lt;0$, and i'm starting to think that splitting it up like that isn't necessary at all. Multiple Inheritance in C# I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic? We moved to a new place where the dryer connection is 3-prong instead of 4-prong. The dryer was originally 4-prong, so we bought a 3-prong cord and installed it. It works fine, except I am not entirely sure if I installed the ground wire correctly.   Originally, there was a green wire on the cord itself, and that was connected to the screw on the top of the first picture. This is, I assume, used to ground the case and prevent shocks when touching the exterior of the dryer.  Now with the 3-pronged plug, there is no ground wire on the cord. So where does the existing green wire in the case go? Do I leave it where it originally was like in the first picture or do I connect it to the central terminal like in the second picture?      There's no bonding strap running from the case to any terminal. I can't tell where the green wire is connected to. How can I get the HTML source of a given web address in C#? Updated WiFi fix for Yoga 13 and Ubuntu 16.04? I have a Lenovo Yoga 13 and I was running 14.04 on it and I've since upgraded to 16.04. I applied the fix descibed here () with 14.04 and it worked but no longer seems to with 16.04. I know that the way Realtek drivers work was reorganized in Linux kernel 4.4 () however I don't know enough about what was changed to know if that was the problem. Has anyone else experienced this issue or know how to fix it? Calling a function of a module by using its name (a string) What is the best way to go about calling a function given a string with the function's name in a Python program.  For example, let's say that I have a module foo, and I have a string whose content is "bar". What is the best way to call foo.bar()?  I need to get the return value of the function, which is why I don't just use eval. I figured out how to do it by using eval to define a temp function that returns the result of that function call, but I'm hoping that there is a more elegant way to do this. We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} &amp; m_{01}&amp;m_{02} \\ m_{10} &amp; m_{11}&amp;m_{12}\\  m_{20} &amp; m_{21}&amp;m_{22}\end{bmatrix}}_{3\times 3}\tag 1 $.   How do I accurately calculate quaternion $q = q_1i+q_2j+q_3k+q_4$for this matrix?Means   how can we write $q_i$s interms of given $m_{ij}$  accurately? I have successfully transferred "Killing Floor" from my brother's laptop to my own via external hard drive. The problem, now that I have all the files, is that I don't know how to launch the game through Steam. All I did was copy files from his Laptop to the hard drive, then from the hard drive to my laptop, then I put the files in the "Steamapps" folder under "Steam" in my laptops hard drive. So the question is, how do I start up the game through Steam? I have an Excel file that has a bunch of VBA and macro code in it.  When I open the file in Excel I can choose not to 'enable' them - so the values in the fields all stay as they were during the last save.  I need to manipulate the values as they were last saved - so I don't want the macros (which look at the current date and update values accordingly) to run.  When I open it via our dot net code:   Dim oxlRep As Excel.Application Dim oWBRep As Excel.Workbook Dim oSheetRep As Excel.Worksheet Dim oRngRep As Excel.Range oxlRep.Open(path)   the vb code runs - throwing off the values.  I've been looking for a way to open it without macros, or in 'secure' mode where the macros aren't run.  If I simply double click the file and don't choose to enable macros the values are all there as I want them.    Usually we run this code within the month that the files are created, so we haven't seen this problem in the 3 or 4 years that it has been working.  Now I need to go back to some of the old files and run some archival code...    Anyone have a suggestion? I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. How to decide on the correct number of clusters? We find the cluster centers and assign points to k different cluster bins in  which is a very well known algorithm and is found almost in every machine learning package on the net. But the missing and most important part in my opinion is the choice of a correct k. What is the best value for it? And, what is meant by best?  I use MATLAB for scientific computing where looking at silhouette plots is given as a way to decide on k . However, I would be more interested in Bayesian approaches. Any suggestions are appreciated. Cannot see files in my external HDD Can I change a user's keyboard input?