wrong result in calculating limit of x*floor(1/x) as x goes to 0
I get the noun form back. Not wrong, but could be better. What were you expecting?
I got the wrong result, namely 0. I attached my file
With Maxima 5.19post I get a noun form too. There have been serveal changes the last time to improve limit. Furthermore, I think the limit of the example is not defined. Therefore, it seems to be not wrong to return a noun form.
Setting the status to pending and works for me.
Isn't the limit 1? Let any x small enough, 1/x = n + e, where n is an integer and e < 1. Then floor(1/x) = n and x*floor(1/x) is n/(n+e) = 1 - e/(n+e). As n gets larger (and x gets smaller), this approaches 1.
Did I make a mistake?
Sorry, I have no mathematical proof. I have come to the conclusion because of
1. The function floor(x) is discontinuous.
2. The function x*floor(1/x) has an infinite number of points of discontinuity
in any infinitesimal intervall when aproching zero.
3. Therefore, the function does not approach a limit.
I could be wrong.
Answer to rtoy. The limit is 1, your proof is essentially orrect. I prefer the following proof: 1/x-1<floor(1/x)<=1/x, hence for
x>0 we have 1-x<xfloor(1/x)<=1. It follows that the limit from the right is 1. The proof for the limit from the left is similar.
Answer to crategus: The function does have a limit, namely 0. Your staement 2. is correct but your conclusion 3. is erroneous.
For the record: Maxima 5.19post gives the result:
That is we get noun form. We expect the answer 1.
Changing the title of this bug report to reflect the problem better and the resolution ID to none.
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