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.
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
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