How to find    $$\lim_{x\to 0}\frac{\sin\tan x-\tan\sin x}{7x^7}?$$   L’Hopital rule does not work because derivative of numerator goes very wide and I need to take 7? Can somebody show me other way of finding it?
Find the limit $$\lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$$  My attempt: Since $$\sin{x}=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\dfrac{1}{7!}x^7+o(x^7)$$ $$\tan{x}=x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+\dfrac{1}{63}x^7+o(x^3)$$ So $$\sin{(\tan{x})}=\tan{x}-\dfrac{1}{3!}(\tan{x})^3+\dfrac{1}{5!}(\tan{x})^5-\dfrac{1}{7!}(\tan{x})^7+o(x^7)$$ Though this method might solve, I think this problem has nicer methods. Thanks.
How to fully delete a git repository created with init? Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ Using $\text{n}^{\text{th}}$ root of unity  $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$  Prove that   $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ Common lines between two files I have the following code that I run on my Terminal.  LC_ALL=C &amp;&amp; grep -F -f  genename2.txt hg38.hgnc.bed &gt; hg38.hgnc.goi.bed   This doesn't give me the common lines between the two files. What am I missing there? Selecting a security team that includes at most one of the oldest man or oldest woman A security company consists of $7$ men and $6$ women. Determine the number of ways to form the night shift team of $3$ men and $3$ women if it can include at most one of the oldest man or oldest woman.  I know the first step is to pick $3$ from each group which is   $_6C_3 \cdot _7C_3 $  but I am not sure how to account for the "include at most one of the oldest man or oldest woman"   The answer is $_6C_3 \cdot _7C_3 + _6C_3 \cdot _5C_3 + _6C_2 \cdot _5C_3 $  Also, how do I account for it in my calculations if I want to include the oldest man and oldest woman from the group? How does Cypher exit/enter the Matrix without an operator? Upper bound on $\chi(G)$ for a triangle-free graph I'm struggling with the following question;     For every graph $G$ such that $K_3 \not\subseteq G$ (i.e. $G$ does not contain a triangle), prove that $\chi(G) \leq 2\sqrt{n} +1$ (where $\chi(G)$ denotes the chromatic number).   Any hints? I really cannot see the connection between the triangle-free nature of $G$ and a chromatic number being bounded above by $O(\sqrt{n})$. Work out the type of c# application a class contained in a DLL is being used by Is there any way to know in C# the type of application that is running.  Windows Service ASP.NET Windows Form Console  I would like to react to the application type but cannot find a way to determine it. If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?   I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.  Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that. Why does parameter expansion with spaces without quotes work inside double brackets "[[" but not inside single brackets "["? I'm confused with using single or double brackets. Look at this code:  dir="/home/mazimi/VirtualBox VMs"  if [[ -d ${dir} ]]; then     echo "yep" fi   It works perfectly although the string contains a space. But when I change it to single bracket:  dir="/home/mazimi/VirtualBox VMs"  if [ -d ${dir} ]; then     echo "yep" fi   It says:  ./script.sh: line 5: [: /home/mazimi/VirtualBox: binary operator expected   When I change it to:  dir="/home/mazimi/VirtualBox VMs"  if [ -d "${dir}" ]; then     echo "yep" fi   It works fine. Can someone explain what is happening? When should I assign double quotes around variables like "${var}" to prevent problems caused by spaces? nginx httpMapModule regex variables The  states that:     Since 1.0.4, case insensitive regular expressions can be used by prefixing the pattern with ~*.   And gives the example:  map $uri $myvalue {     /aa                   /mapped_aa;     ~^/aa/(?&lt;suffix&gt;.*)$  $suffix; }   I updated to nginx/1.2.7 to make use of this, and naive regexes work okay, but variables don't. When I try the above syntax I get the error:  nginx: [emerg] pcre_compile() failed: unrecognized character after (?&lt; in "^/aa/(?&lt;suffix&gt;.*)$" at "suffix&gt;.*)$"   I also tried the following syntaxes, but none of them worked:  ~^(.*)$  $1; ~^(.*)$  \1;   but neither of them worked either.  Does anyone know how to get variables to work in regular expressions in HttpMapModule? What does the notation like 8.6e-28 mean? What is the 'e' for? Is hyperref really the best way to add metadata to a TeX file? So I know one of hyperref's features is that it can add metadata to you PDFs. This is a nice feature and if you're putting stuff online, it's good to include metadata.  But is the hyperref package really the best way to do it? I mean, if you're using hyperref anyway, that's fine. But if you don't want hyperrefs in your document, it seems weird to include the package, and then turn off pretty much all its functionality. And given that hyperref is a shorthand for "package conflict nightmare", is seems insane that you should have to deal with it just for the sake of some metadata.  So I have two questions.   Is there a good reason that hyperref includes this functionality which is apparently unrelated to its primary purpose? Is there a better (smaller, less conflicty, easier...) option to include metadata in my PDFs? How to parse hundred html source code files in shell? How do I find the index of an element in a list in Racket? Why $SU(3)$ and not $U(3)$? This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress.  I'd like to prove that any linear functional $\phi$ on $\ell^\infty$ that is positive, i.e. for any $(x_n) \in \ell^\infty$ such that $x_n \geq 0$ for all $n \in \mathbb{N}$ we have $\phi((x_n)) \geq 0$, is a bounded linear operator and has $\|\phi \| = \phi((1,1,...))$.  Nothing I have tried worked so far, so any hints would be appreciated, thanks! There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint? I'm trying modeling things like this using oracle:  An account can have 1~5 members  So I'll have a account table and a member table with a FK of account. But is that possible to check if the account have less than 5 members?  I've tried constraint and trigger but oracle doesn't allow sub queries in constraint or condition of trigger like  CHECK( SELECT ...) or IF count(*) &gt;= 5  which make things getting hard. I have an Asus N76VB notebook with Ubuntu 14.04 and TLP for power managment. However, the fan is always on. It's not really loud, but it's annoying. My CPU usage is around 1-2%. It's clocked at 1.20 GHz.  I've added acpi=force to my boot parameters but it doesn't change anything.  /proc/acpi/fan: No such file or directory /proc/acpi/thermal_zone: No such file or directory   I've loaded all Asus ACPI modules.  lsmod | grep asus:  asus_nb_wmi            16990  0  asus_wmi               24191  1 asus_nb_wmi sparse_keymap          13948  1 asus_wmi video                  19476  2 i915,asus_wmi wmi                    19177  2 mxm_wmi,asus_wmi   Somehow, it can read the temperatures.  sensors:  acpitz-virtual-0 Adapter: Virtual device temp1:        +43.0°C  (crit = +108.0°C)  coretemp-isa-0000 Adapter: ISA adapter Physical id 0:  +44.0°C  (high = +87.0°C, crit = +105.0°C) Core 0:         +44.0°C  (high = +87.0°C, crit = +105.0°C) Core 1:         +42.0°C  (high = +87.0°C, crit = +105.0°C) Core 2:         +39.0°C  (high = +87.0°C, crit = +105.0°C) Core 3:         +39.0°C  (high = +87.0°C, crit = +105.0°C)  asus-isa-0000 Adapter: ISA adapter temp1:        +43.0°C     I really hope someone knows how to solve this. Almost all information I could find was about saving power to create less heat.    I got Intel and NVIDIA (GeForce 740M) graphics but I'm almost anytime on Intel.  $ sudo lshw -c display   *-display                       description: VGA compatible controller        product: 3rd Gen Core processor Graphics Controller        vendor: Intel Corporation        physical id: 2        bus info: pci@0000:00:02.0        version: 09        width: 64 bits        clock: 33MHz        capabilities: msi pm vga_controller bus_master cap_list rom        configuration: driver=i915 latency=0        resources: irq:43 memory:f7400000-f77fffff memory:d0000000-dfffffff ioport:f000(size=64)     fwts fan: Why does the Hilbert curve fill the whole square?