Re: Proving the existence of a limit



On Fri, 8 Sep 2006, Paul Smith wrote:

Even if you exclude that line from the domain, the function takes on
arbitrarily large values in every neighborhood of (0,0).

Thanks, Dave, but how can one prove that?

Paul

Once again, it was V(x,y) = x*y/(y-x)^(1/2) near (0,0).

It is well-defined on the halfplane y>x, and (0,0) is an accumulation
point - so far so good.

I did the arithmetic, and here we go: for every k>0 (to be on the safe
side), the curve given by

y = x + x^4 / (k + (k^2 - x^3))^2

defined for 0 < x < k^(2/3), lies in that halfplane, has (0,0) as an
accumulation point, and the value of the expression V(x,y) along the curve
is up to you to calculate. (Experiment first, then simplify.) So, why does
V(x,y) fail to have a limit, big time?

And: How did I discover the curve???

Cheers, ZVK(Slavek).
.

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