Re: Cantor Confusion

Virgil wrote:

In article <1166090594.020341.42340@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

I do not understand why you argue that in
lim{x -> oo} lim{y -> oo} (2x + 3y)/xy = 0 = lim{y -> oo} lim{x -> oo}
(2x + 3y)/xy interchanging limits is not possible.

Nor do I.

But for lim{x -> oo} lim{y -> oo} (2x + 3y)/(x + y) = 3
and lim{y -> oo} lim{x -> oo} (2x + 3y)/(x + y) = 2,
one cannot exchange the order of the limits without changing the value of the result.

This is highly misleading. What's really interesting is the behaviour of
the function f(x,y) = (2x + 3y)/(x + y) for large values of x _and_ y.

This can be investigated by substituting x = t.cos(phi) , y = t.sin(phi)
Giving: f(phi) = (2.cos(phi) + 3.sin(phi))/(cos(phi) + sin(phi)) which
turns out to be independent of t for t <> 0 . So for (x,y) -> oo we find
a much more complicated behaviour, dependent on the angle phi, being the
the direction in which we are "looking". It turns out that the function
has several singularities, namely where sin(phi) + cos(phi) = 0, and, as
a function of phi, it is increasing everywhere; take e.g. the derivative
f'(phi) = 1/(cos(phi) + sin(phi))^2 . For phi = 0 we find indeed a limit
= 2 and for phi = pi/2 we find indeed a limit = 3 , but these limits are
only quite special cases of the function's "real" behaviour at (oo,oo).

Han de Bruijn

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