Verify Multivariable Limits w/ Delta-Epsilon Arguments

In summary, the conversation discusses using delta-epsilon arguments to verify a limit of xy^2 = 1 as (x,y) approaches (1,-1). The individual is struggling with simplifying the equation and utilizing the basic ideas of delta-epsilon proofs. A hint is given to write the equation in a different form in order to solve it.
  • #1
Volt
10
0

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?
 
Physics news on Phys.org
  • #2
Hint: Try writing |xy2 -1| = |(xy2 - y2) + (y2 - 1)| to get started.
 

FAQ: Verify Multivariable Limits w/ Delta-Epsilon Arguments

What is a multivariable limit?

A multivariable limit is a mathematical concept that represents the behavior of a function as the input values approach a specific point in a multi-dimensional space. It determines the value that a function approaches as the input values get closer and closer to the specified point.

What is the purpose of verifying multivariable limits with delta-epsilon arguments?

The purpose of using delta-epsilon arguments to verify multivariable limits is to rigorously prove that a limit exists and determine its value. This method involves finding a suitable value for delta (distance from the specified point) and epsilon (tolerance around the limit) and showing that for any value of delta, there exists an epsilon such that the function values are within that tolerance from the limit.

What are the steps for verifying multivariable limits with delta-epsilon arguments?

The steps for verifying multivariable limits with delta-epsilon arguments are as follows:1. Write the limit expression in terms of delta and epsilon.2. Determine a suitable value for delta based on the distance from the specified point.3. Show that for any value of delta, there exists an epsilon such that the function values are within that tolerance from the limit.4. Use algebraic manipulation to show that the function values can be made to satisfy the epsilon bound.

What are the common challenges in verifying multivariable limits with delta-epsilon arguments?

Some common challenges in verifying multivariable limits with delta-epsilon arguments include:1. Difficulties in finding a suitable value for delta.2. Complex algebraic manipulations.3. The need for a strong understanding of limits and the properties of functions.4. Limited applicability to certain types of functions, such as piecewise functions.

Why is it important to verify multivariable limits with delta-epsilon arguments?

Verifying multivariable limits with delta-epsilon arguments is important because it provides a rigorous and precise way to determine the existence and value of a limit. It also helps in understanding the behavior of a function in a multi-dimensional space, which has many applications in fields such as physics, engineering, and economics.

Similar threads

Back
Top