Re: Double limits at the infinity.
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 21 Jul 2005 21:16:27 GMT
In article <dbo1uj$ph0$1@xxxxxxxxxxxxxx>,
Jesús Cid-Sueiro <jcid@xxxxxxxxxxx> wrote:
>Given function f(x,y) I would need to know under which conditions over
>f, the limit of f as x,y go to +infinity does not depend on the way to
>go, for instance
>
>lim_{y-->inf} lim_{x-->inf) f(x,y) =
>lim_{x-->inf) lim_{y-->inf} f(x,y) =
>lim_{x-->inf} f(x,x) =
A sufficient condition is that L = lim_{(x,y) -> (infty,infty)} f(x,y)
exists, i.e. for every epsilon > 0 there is N such that whenever x > N
and y > N, |f(x,y) - L| < epsilon.
>I guess that the above relations are true if the limit exists, f is
>continuous and df(x,y)/dx and df/dy always go to zero as x and y go to
>+inf (in any way).
>Is this correct?
No (if by "the limit" you mean the three limits you mentioned).
Consider, say, f(x,y) = arctan((x-y)/sqrt(x+y)).
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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