Prove $\lim \frac{cos(x)-1}{x} = 0$ without l'Hopital
Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$
What is the rigor behind U subsitution? How to change the size of desktop icons? I tried gconf-editor as given at  .  I am not getting the option for changing size at app &gt; nautilus &gt; icon_view.  I am using Ubuntu 11.10. Kindly help me in changing default icon size.   I want to make them small. Why is dark matter the best theory available to explain missing mass problems? Why is dark matter the best theory to explain the missing mass problem?    Why is dark matter mathematically necessary to explain the missing mass problem?    On a side not I believe dark matter is definately intuitive the next logical step in physics is to accept the fact the direct observations can not always be the determining factor in the acceptance of a theory.  Positivism is not going to be the only driving force in physics forevermore.    One theory that seems almost as good is the supersymmetric particle. What determination which are actually detectible and how would we distinguish the differences between the two?    References:   . Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\mathrm dx$ Calculate the definite integral      $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;\mathrm dx $$      given that $a&gt;b&gt;0$.   My Attempt:  If we replace $x$ by $C$, then  $$ I = \int_{0}^{\pi}\frac{\sin^2 C}{a^2+b^2-2ab\cos C}\;\mathrm dC $$  Now we can use the Cosine Formula ($A+B+C=\pi$). Applying the formula gives  $$ \begin{align} \cos C &amp;= \frac{a^2+b^2-c^2}{2ab}\\ a^2+b^2-2ab\cos C &amp;= c^2 \end{align} $$  From here we can use the formula $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ to transform the integral to  $$ \begin{align} I &amp;= \int_{0}^{\pi}\frac{\sin^2 C}{c^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2A}{a^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2 B}{b^2}\;\mathrm dC \end{align} $$  Is my process right? If not, how can I calculate the above integral? I need to simulate the warning:  @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @ WARNING: REMOTE HOST IDENTIFICATION HAS CHANGED!  @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@  IT IS POSSIBLE THAT SOMEONE IS DOING SOMETHING NASTY!   I have two Linux machines and when I do ssh from Linux1 to Linux2, ssh accepted and reports no problem.  But because some of testing, I want to simulate the - Identification Has Changed error  So what I need to edit on the remote server in order to create the problem?  what I did until now is to change the knpwn_hosts on both machine but its not creating the problem.  I get this:   key_read: uudecode AAAAB3NzaC1y3jrzG62xyHrb3qxc9eYpg1Cp1PrRElTFfYbEHsRfTZU21S8kO5/Tq+ED8=  failed  key_read: uudecode AAAAB3NzaC1y3jrzG62xyHrb3qxc9eYpg1Cp1PrRElTFfYbEHsRfTZU21S8kO5/Tq+ED8=  failed  The authenticity of host 'ste1 (10.106.164.111)' can't be established.  RSA key fingerprint is 84:82:2b:80:ca:3f:57:c7:8c:14:44:dd:b9:70:e5:d8.  Are you sure you want to continue connecting (yes/no)? Installed Teamviewer using a 64-bit system, but I get a dependency error Battery Ampere-hour rating vs Battery Amps (not an experienced user) I have almost no experience whatsoever with the technicals of electronics so this is probably a very easy question for someone who does.  Do you know if a battery with a 200Ah rating can put out 200A for one hour or are there limitations? According to this website's third paragraph (), you can.     For example, an average automotive battery might have a capacity of about 70 amp-hours, specified at a current of 3.5 amps. This means that the amount of time this battery could continuously supply a current of 3.5 amps to a load would be 20 hours (70 amp-hours / 3.5 amps). But let’s suppose that a lower-resistance load were connected to that battery, drawing 70 amps continuously. Our amp-hour equation tells us that the battery should hold out for exactly 1 hour (70 amp-hours / 70 amps), but this might not be true in real life. With higher currents, the battery will dissipate more heat across its internal resistance, which has the effect of altering the chemical reactions taking place within. Chances are, the battery would fully discharge some time before the calculated time of 1 hour under this greater load.   But wouldn't that mean you could hook up a 200Ah battery and ask it to put out 12000A for one minute or 720000A for a second?  That seems very unrealistic, lol.  I'm trying to find the proper kind of off grid battery that can power a microwave through a 3000 watt power inverter.  The microwave needs 1800 watts and the battery needs to be 12 volts so that would mean I need about 150 amps, I'm wondering if a battery with a 200Ah rating could do it? How to stop apt from doing anything Automatically updating column in feature class when another column is changed? I have a feature class in a file geodatabase and it contains a field called [curing] in short integer type and another field called [IsPopulated] in short integer as well.  What I want to achieve is to make [IsPopulated] automatically populated with "0" or "1" depending on whether the [curing] column is filled with a numeric value.  "Field Calculator" does not help as it's just a one-off operation and NOT automatic. Say, for example, you and a group of people were all sailing out in the ocean and something happens...then you say,  "I guess we are all in the same boat"  You are literally in the same boat with everyone else, and your current situation is the same as everyone else.  Now such a phrase can be applied literally and figuratively. I was wondering if there was a word that would mean both literally and figuratively. I know this is a tad bit of an oxymoron, but I was curious to see if there was such an "all encompassing" word for this. How can I make something that looks like this?  --------   Text   ---------  where the horizontal rules extend to the margin on either side? Extended Rigid Bodies in Special Relativity I was reading Landau &amp; Lifshitz's Classical Theory of Fields and I noticed that they mention that an extended rigid body isn't "relativistically correct".   For example, if you consider a rigid rod and apply a torque at one end, by definition of being rigid, the whole body must start rotating at the same instant. But the information about the force being applied cannot travel faster than the speed of light.   The book hasn't mentioned how to resolve this paradox, and after thinking about it for a while, I'm wondering if one can even define a rigid body in a relativistically correct manner.  To summarize: How does one define a rigid body in special relativity? Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated. I want to add this iostat command to my .bashrc and create an alias for it: iostat -xk 2 $(findmnt -T ~ | awk 'END {print $2}')  I'm adding this to my .bashrc: alias ios='iostat -xk 2 $(findmnt -T ~ | awk 'END {print $2}')'  Unfortunately, it doesn't work. If I run the above iostat command in a terminal it works, but when I run the ios alias it no longer works. And yes, I restart my shell every time. Running out of inodes I just installed a fresh install of 32-bit 12.04 LTS Pangolin today. When I tried to install a few applications, it was really slow. But when I download a .deb or .tar.gz file from on my web browser, it is a lot faster.  Computer specs:  HP Pavilion Slimline s7700n PC  [Memory]         873.7 MiB  [Processor]      AMD Athlon(tm) 64 X2 Dual Core Provessor 3800+ x 2  [Graphics]       GeForce 6105 LE/integrated/SSE2/3DNOW!  [Disk Size] 638.5 GB  I normally get 200-400 kbps.  This is the comparison. Overview of basic facts about Cauchy functional equation The  asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman classes to constructing useful counterexamples for some advanced questions. Solutions of this equation are often called . Also a few other equations related to this equation are often studied. (Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.) Is there some overview of basic facts about Cauchy equation and related functional equations - preferably available online? I recall from complex analysis a while ago that the series $$\sum_{n=1}^\infty \frac{z^{n!}}{n}$$ has radius of convergence 1, diverges on roots of unity, and converges elsewhere on the circle. It's simple enough to see that it has to diverge on roots of unity: consider $z$ such that $z^k = 1$ for some $k \in \mathbb{N}.$ Plug this into the series and you see that past the $k$'th term, $z^{n!} = (z^k)^m = 1$ for some $m \in \mathbb{N}$; hence, the series becomes harmonic and must diverge.  But I don't recall how to show that the series converges for other values on the circle. If this is simple I'd like just a hint, if it is involved I would prefer a sparse outline of the proof. Thank you! "To which", "by which", "on which" etc I have come across the phrases like "to which","for which", "by which", "on which" and so on(using a preposition with a relative pronoun).   e.g. The chair on which the body was found..  Could someone please explain if there's a particular rule to be followed when building/using them?   Can I use any preposition before 'which'?  What if I say "The chair which the body was found on"? How would I decide which preposition to be put? Can I use other relative pronouns too?  Thanks in advance. How does Spider-Man climb walls?