Using the Delta-epsilon definition but for two variables?

In summary, the conversation is discussing the formal definition of a limit in multiple dimensions and how it can be extended using epsilon-delta notation. The limit in question is for the function f(x,y)= y / (x^2 + 1) with an epsilon value of 0.05. The conversation also mentions using single variable analysis to make the expression small enough.
  • #1
pr0me7heu2
14
2
I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05
 
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  • #2
For multiple dimensions you need to extend the epsilon-delta notation to incorporate the epsilon-neighborhood of the point p0 = (x,y), which is the set of all points p such that the distance between p0 and p is less than epsilon.
 
  • #3
pr0me7heu2 said:
I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05

The limit seems to be 0.
So if [tex]0<|y|<\delta[/tex] and [tex]0<|x|<\delta[/tex].

[tex]\left| \frac{y}{x^2+1} \right| \leq \frac{|y|}{x^2} < \frac{\delta}{x^2}[/tex].

Now use single variable analysis that [tex]\frac{\delta}{x^2}[/tex] can be made sufficiently small.
 
  • #4
pr0me7heu2 said:
I have probably over thought the whole thing... but I can't seem to find any place to start with this one:

Using the formal definition of a limit:

f(x,y)= y / (x^2 + 1) e=0.05

limit as (x,y) goes to what?
 

1. What is the Delta-epsilon definition for two variables?

The Delta-epsilon definition for two variables is a mathematical concept used to determine the limit of a function as two variables approach specific values. It is based on the idea that the function's output will get arbitrarily close to a certain value as the variables get closer and closer to specific values.

2. How is the Delta-epsilon definition for two variables different from the single variable definition?

The single variable Delta-epsilon definition only considers one variable approaching a specific value, while the two variable definition takes into account both variables approaching specific values simultaneously. This allows for a more precise determination of the limit of the function.

3. When is the Delta-epsilon definition for two variables used?

This definition is commonly used in multivariable calculus and other advanced mathematics courses to determine the limit of functions with two or more variables. It is also used in real-world applications, such as in physics and engineering, to model the behavior of systems with multiple variables.

4. What are the steps for using the Delta-epsilon definition for two variables?

The steps for using this definition are similar to the single variable case. First, we must state the definition and the values the variables are approaching. Then, we must show that the function's output gets arbitrarily close to the limit value as the variables approach the given values. This is typically done by manipulating the delta and epsilon values and showing that they are within a certain range of each other. Finally, we must conclude that the limit of the function exists and equals the given value.

5. What are some common mistakes made when using the Delta-epsilon definition for two variables?

One common mistake is not considering the approach of both variables simultaneously. It is important to show that the function's output gets arbitrarily close to the limit value as both variables approach the given values. Another mistake is not manipulating the delta and epsilon values properly, which can lead to incorrect conclusions about the limit of the function. It is also important to carefully state the definition and values being used to avoid confusion.

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