Finding The Limit Of A 3D Function

In summary: So the limit becomeslim_{r\to0} r^2\cos(\theta)\sin(\theta)/[r^2\cos^2(\theta)+r^4\sin^4(\theta)]=lim_{r\to0} \cos(\theta)\sin(\theta)/[\cos^2(\theta)+r^2\sin^4(\theta)]=0/[\cos^2(\theta)]=0In summary, the limit as (x,y) approaches (0,0) of (x^2y)/(x^2+y^4) is equal to 0, as
  • #1
Baumer8993
46
0

Homework Statement


Find the limit as X, and Y both approach 0 of (X2Y)/(X2 + Y4)


Homework Equations


The equation from above.


The Attempt at a Solution


I have been doing the technique of approaching from different lines such as y = x, or x = y,
x = 0, and y = 0. All of them give me a limit of zero, so that will not do. I graphed the function online, and it appears as if the function does have a limit of zero. How would I prove this? I think I need to do something with polar coordinates, but I am not sure how to do that.
 
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  • #2
EDIT:
I'd go with SammyS's
 
  • #3
Baumer8993 said:

Homework Statement


Find the limit as X, and Y both approach 0 of (X2Y)/(X2 + Y4)

Homework Equations


The equation from above.

The Attempt at a Solution


I have been doing the technique of approaching from different lines such as y = x, or x = y,
x = 0, and y = 0. All of them give me a limit of zero, so that will not do. I graphed the function online, and it appears as if the function does have a limit of zero. How would I prove this? I think I need to do something with polar coordinates, but I am not sure how to do that.
Yes. Change to polar coordinates.

[itex]x=r\cos(\theta)[/itex]

[itex]y=r\sin(\theta)[/itex]
 
Last edited:

1. What is the purpose of finding the limit of a 3D function?

The purpose of finding the limit of a 3D function is to determine the behavior of the function as it approaches a specific point or value in 3D space. This helps us understand the behavior of the function and make predictions about its values at that point.

2. How is the limit of a 3D function different from the limit of a 2D function?

The limit of a 3D function is similar to the limit of a 2D function in that it represents the value that the function approaches as the input approaches a certain point. However, in 3D space, the input is approaching a specific point in three dimensions, instead of just two, making the calculation more complex.

3. What techniques can be used to find the limit of a 3D function?

Some techniques that can be used to find the limit of a 3D function include using algebraic manipulation, graphing the function in 3D space, and using the limit laws and theorems to simplify the calculation. Computing the limit using a computer program or calculator is also a common method.

4. Can the limit of a 3D function be undefined?

Yes, the limit of a 3D function can be undefined. This can occur when the function has a discontinuity or a point where the values approach different values from different directions. It can also be undefined if the function is undefined at the point where the limit is being calculated.

5. How can finding the limit of a 3D function be applied in real-world situations?

Finding the limit of a 3D function has many real-world applications, such as in physics, engineering, and economics. For example, it can be used to determine the maximum or minimum values of a function, which can help in optimizing designs or predicting the behavior of a system. It can also be used to analyze the behavior of functions in 3D space, such as the path of a projectile or the surface area of a three-dimensional object.

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