Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Limits problem in two variables

let

if x ≠ 0 and y ≠ 0, and otherwise

prove that:

exists, but that the iterated limits:

and

do not exist.

I don't get it. I get 2 for all of them!

Also, I notice this function defines f(x,y) = 0 if x or y = 0, but since we're dealing with limits approaching zero, don't we not really need to know what happens at exactly 0?

Last edited by mikau (2008-09-22 14:22:47)

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Limits problem in two variables

I think you're getting 2 by thinking of

.

This is different though. As x approaches 0, sin x oscillates more and more rapidly and doesn't have a limit. However, as it's a sine function, it's bounded by ±1 and so x sin(1/x) will be bounded by ± x.
x going to 0 therefore means that the limit is 0.

You're right that we don't need to know what happens exactly at 0, I'm guessing the book (or who/whatever set the question) just wanted the function to be well-defined for all reals.

(Also I'd have thought that those iterated limits do exist, so I can't help you there. )

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It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Limits problem in two variables

I agree mathsyperson, they do exist.  If we had:

Then they both wouldn't... but neither would the limit (x, y) -> (0,0)

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