So just as the question states, I'm wondering how python calculates equations such as   def indeterminate(x):     return (x - math.sin(x)) / math.pow(x, 3)   where Limit approaches 0. I'm asking because when I input   &gt;&gt;&gt; indeterminate(0.0000001) 0.17205356741102978   Where as   &gt;&gt;&gt; indeterminate(0.00000001) 0.0   Shouldn't the output be approaching 1/6? And if it becomes too close, wouldn't a ZeroDivisionError message be more suitable than 0.0?  Another question I have is how do I calculate the actual number of   indeterminate(0.00000001)   The correct answer using L'hopital's rule should be 1/6 but (0.166666...) and not 0.1720...
Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen?
Positioning of decorations on parallel lines I am running out of storage, I usually have bought a lot of my games on disk but if I buy a game on the store then delete it, and want to redownload it do I have to buy it again How to draw good arrows explaining parts of equation? Proving replacement theorem? I want to see if I am understanding the proof of the replacement theorem correctly.   Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors.  Let $L \subseteq V$ be a linearly independent subset containing $m$ vectors.  Then $m\leq n$ and there exists a subset H of G containing exactly $n-m$ vectors s.t. $L \cup H$ generates $V$.   The proof in the book says to start with $m=0$. In that case $L=\varnothing$, the empty set.        Let's assume this theorem is true for some integer $m \geq 0$.   Let $L = \{v_j\}_{j=1}^{m+1}$ and define it as a linearly independent subset of $V$ consisting of $m+1$ vectors.  Since any subset of a linearly independent set is linearly independent as well ($S_1 \subseteq S_2 \subseteq V$), then $\{v_j\}_{j=1}^m$ is linearly independent also.   It then says to use the induction hypothesis to say that $m \geq n$. Well, ok, but that seems sort of pulled out of nowhere to me. The next step is to say that there is therefore another subset, $\{u_k\}_{k=1}^{n-m}$ of G such that $\{v_j\}_{j=1}^m \cup \{u_k\}_{k=1}^{n-m}$ spans $V$.   That being the case there are scalars $(a_j)_{j=1}^m$ and $(b_k)_{k=1}^{n-m}$ which you can multiply by the vectors $v_j$ and $u_k$. You can then add the two sets of vectors, yielding $$\sum_{j=1}^m a_j v_j + \sum_{k=1}^{n-m} b_k u_k = v_{m+1}$$  At this point I am not sure I understand things. Because we are assuming $n-m&gt;0$ -- otherwise $v_{m+1}$ is linearly dependent. But then it says not only is $n&gt;m $ but $n&gt;m+1$, and that is where I am losing the plot to get to the next step. bst file not found Punctuating a quoted question within a quoted statement I'm working on a story, and I find myself faced with something of a quandary.  Specifically, I need to know the proper style for the following:     “I have thousands of them,” the commissioner answered, “but I’ll start with ‘What do you need?’.”   Is it proper form to have both the question mark and period, or should one of them be dropped? Copy a file back to local system with ssh If I'm logged in to a system via SSH, is there a way to copy a file back to my local system without firing up another terminal or screen session and doing scp or something similar or without doing SSH from the remote system back to the local system? Incorrect bookmarks and page number in table of contents I have problem with table of contents and with bookmarks. In pdf file   table of contents shows wrong page number of Introduction and Bibliography, Bookmarks: introduction and bibliography don't bring me to the correct page.   Part of my code  \documentclass[12pt,a4paper]{book}  \usepackage[polish]{babel} \usepackage{polski} \usepackage[cp1250]{inputenc} \usepackage[pdftex,pagebackref=false,draft=false,pdfpagelabels=false,pdfstartview=FitH,pdfstartpage=1,bookmarks=true,pdfauthor={LiN},pdftitle={PD},pdfsubject={Titlei},pdfkeywords={words},unicode=true]{hyperref}  \usepackage{tocloft}  \begin{document} \tableofcontents \addcontentsline{toc}{chapter}{Introduction} \chapter*{Introduction} \chapter{Chapter1} \section{Section} \addcontentsline{toc}{chapter}{Bibliography} \begin{thebibliography}{9999999}% \bibitem{1}Author, \emph{Title of book}, Publisher. \end{thebibliography} \end{document}   How can I change that? Why does magnetic force vary in proportion to the cube of the distance instead of the square? Magnetic forces vary from gravity and electromagnetic radiation (such as light) in that gravity and radiation diminish by the square of the distance from the source, but magnetism diminishes by the cube of the distance.  Why is this? The set of real numbers and power set of the natural numbers I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? Does leaning (banking) help cause turning on a bicycle? I think it's clear enough that if you turn your bicycle's steering wheel left, while moving, and you don't lean left, the bike will fall over (to the right) as you turn. I figure this is because the bike's momentum keeps it moving in the direction you were going, and since your wheels have friction against the ground, the top of the bike moves forward relative to the bottom of the wheels. The top of the bike going north while the bottom of the wheels go northwest will understandably cause you to topple. So to counteract this and keep you from falling over, leaning into the turn is necessary. But is there also a direct causal relationship -- that leaning will cause the bike to start to turn? If I start leaning left, I will turn left... but maybe that's because I know that if I don't turn the steering wheel left, the bike will fall over (to the left). I experimented with unruly turns of the steering wheel when I was a kid, and got my scrapes and bruises. Now that I'm a cautious and sedate adult I'm not anxious to experiment that way. :-) (I also want to ask why airplanes bank into a turn... they don't have the same issues as a bike, i.e. the bottom part has no special friction against the ground. But that would probably make the question too broad.) I never dealt with Android devices before so I'm completely newbie at it, please comment if something is too simple.  Say I purchase , that has got Android Froyo 2.2 installed in it.  If there is a new version of Android released tomorrow or next month, will I be able to update it (like I will be able to update my iPod Touch 4 to iOS 5)? Invocable method Apex class is not visible in Process builder Is there any way to see the timetable for a  bus route?  On their website I can see the routes map, but I'd me more interested in seeing where the bus is going to stop for a given route.  E.g. if a bus goes from Rotterdam to Amsterdam, I'd like to see each stop. Legendre symbol: Showing that $\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0$ I have a question about Legendre symbol.  Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies: $$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$  I know that $\displaystyle\sum_{m=0}^{p-1} \left(\frac{m}{p}\right)=0$, but how do I connect this with the previous formula?  Any help is appreciated. Respawning the Ender Dragon Which mathematical operation does the right hand rule for current come from? I am currently wondering about the famous .  Where does it come from mathematically that when you point with your thumb in the direction of the current, your curved fingers will point in the direction of the $B$-field? In other words, which mathematical operation does it come from? Probably it has to do with some vector product or Stokes theorem application, but I am not quite sure about it. Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not? Does the force of kinetic friction increase with the relative speed of the objects involved?    I have heard and read that the answer is no.  This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting.  What phenomena are at work with kinetic friction, and why does it not increase with relative speed? I've been hearing some rumors from friends, mostly anecdotal indicating that skills, when gone past 10 (such as initially having 10 in one skill, then finding a bobblehead or skillbook), actually improve past 10.     I can't find any proof of this yet, and my current character right now is slightly incapable of maxing any stats soon to 10 and testing it out. So I'd like to understand if SPECIAL stats increased past 10 show any benefit?  Can SPECIAL stats be increased past 10? Word for a temporary project name