Multivariable limits

  • Thread starter Volt
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Homework Statement



Verify the following limit by using delta-epsilon arguments

Homework Equations



lim (x, y) -> (1, -1) of xy^2 = 1

The Attempt at a Solution



Right, so I'm having some trouble with these delta-epsilon proofs for multivariable limits. Some of them are easier than others; I'm talking mainly about cases where the limit = 0, when the limit is some other constant like in the above question I'm not sure how to simplify it and try to get an answer.

I'm aware of some of the basic ideas here, like:

sqrt(x^2 + y^2) < delta => |f(x,y) - L| < epsilon

This also implies that |x| < delta and |y| < delta, which seems to be what you use in practice to solve most of these things, rather than the above definition.

If I try that on the above equation though, I get |delta^3 - 1| < epsilon. What on earth do I do from here? How do I deal with the constant in situations like these?
 

Answers and Replies

  • #2
LCKurtz
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Hint: Try writing |xy2 -1| = |(xy2 - y2) + (y2 - 1)| to get started.
 

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