On 11 January 2012 21:44, Joachim Durchholz <[email protected]> wrote:
> Am 11.01.2012 11:32, schrieb Alexey U. Gudchenko: > >> 11.01.2012 12:51, Joachim Durchholz пишет: >> >>> x²/x has a discontinuity for x=0, and x does not, hence x²/x is not the >>> same as x. >>> >> >> No, no, it is continuous because the limit when x-->0 exists (equals 0), >> > > The limit exists, but that's just half of the definition of "continuous". > > > > and the same as a value of function at this point, 0**2/0 (which by > >> definition is equal 0). >> > > f(x) = x²/x has no definition for x=0. It involves division by zero. > > If you go from functions to relations, then for a, b != 0, we have > - a/b is a one-element set > - a/0 is the empty set since no r satisfies 0*r = a > - 0/0 is the the set of all values since all r satisfy 0*r = 0 > So you can assign an arbitrary value to the result of 0/0 and it will be > "correct", but you don't know whether the value you assigned is "more" or > "less" correct than any other. > For example, what should (x²/x)/(x²/x) be for x=0? > If you do the x-->0 limit first, you'll get 1. > If you stick with the basic substitutability rules of math, you get > (x²/x)/(x²/x) = (0)/(0) = 0/0 = 0. > If you stick to the basic substitution rules you completely ignore the way this behaves *around but not at* zero. And 0/0 is not 0. It's undefined. > Hilarity ensues. > > Oh, and I bet different people will have different assumptions about which > rule should take priority. > And if their assumptions differ from those that simplify() applies, they > will come here and ask what's wrong. > > Regards, > Jo > > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to sympy+unsubscribe@** > googlegroups.com <sympy%[email protected]>. > For more options, visit this group at http://groups.google.com/** > group/sympy?hl=en <http://groups.google.com/group/sympy?hl=en>. > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
