[Limits] Help with Delta-Epsilon Proofs for Multivariable Functions

In summary, a delta-epsilon proof is a mathematical technique used to prove the limit of a function. For multivariable functions, it involves choosing values for delta in each variable and showing that for any value of epsilon, there exists a corresponding value of delta such that the difference between the function and its limit is less than epsilon for all values within delta of the limit. The key steps in this proof are choosing values for delta, setting up an inequality, simplifying the inequality, and choosing a value for delta that satisfies the inequality for any value of epsilon. Delta-epsilon proofs are important in mathematics as they provide a rigorous method for proving the existence of a limit. However, challenges and common mistakes in this type of proof include choosing appropriate
  • #1
Steve1231
1
0
Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions.
here is my question:

Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists.
The part that has me confused is the y to the power of 3, where as most of the examples I've worked through thus far only contain squared variables.
Any help is appreciated =)
 
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  • #2
The "y^2 + 3y^2" in the denominator seems odd. Are you sure the problem is typed properly?
 

FAQ: [Limits] Help with Delta-Epsilon Proofs for Multivariable Functions

What is a delta-epsilon proof?

A delta-epsilon proof is a mathematical technique used to prove the limit of a function. It involves choosing a value for delta (denoted as ∆) and showing that for any value of epsilon (denoted as ε), there exists a corresponding value of delta such that the difference between the function and its limit is less than epsilon for all values within delta of the limit.

How does a delta-epsilon proof work for multivariable functions?

In the case of multivariable functions, a delta-epsilon proof involves choosing a value for delta in each variable and showing that for any value of epsilon, there exists a corresponding value of delta such that the difference between the function and its limit is less than epsilon for all values within delta of the limit. This process is repeated for each variable in the function.

What are the key steps in a delta-epsilon proof for multivariable functions?

The key steps in a delta-epsilon proof for multivariable functions are: choosing a value for delta in each variable, setting up an inequality using the distance formula to represent the difference between the function and its limit, simplifying the inequality, and choosing a value for delta that satisfies the inequality for any value of epsilon.

What is the importance of delta-epsilon proofs in mathematics?

Delta-epsilon proofs are important in mathematics because they provide a rigorous method for proving the existence of a limit. This allows for precise understanding and analysis of the behavior of functions, which is essential in many areas of mathematics and science.

Are there any challenges or common mistakes in delta-epsilon proofs for multivariable functions?

One common challenge in delta-epsilon proofs for multivariable functions is choosing appropriate values for delta in each variable that satisfy the inequality for any value of epsilon. This requires careful analysis and understanding of the function and its limit. Additionally, common mistakes include not simplifying the inequality properly, or not following the order of operations correctly.

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