Multi-variable limit using epsilon-delta technique

In summary, the conversation is about a mechanical engineering student struggling with proving the limit of a multivariable equation using the epsilon-delta technique. The student is unsure of how to use this technique and the book does not provide clear instructions. Another person suggests rewriting the function in polar coordinates to make it easier to see that the limit goes to zero as the radius approaches zero.
  • #1
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Hi guys, I'm new to this forum, I am a mechanical engineering student, I am actually taking Multi-variable Calculus (Calculus III, on my college) this semester and I'm having a lot of trouble proving the limit of a multivariable equation using the epsilon-delta technique.

Example:

lim (3xy^2)/(x^2+y^2) x->(0,0)

I don't even understand what is about, I look the limit of the equation along x and y-axis y=mx and y=x^2 and found L=0 but how do I prove it is really 0.
The book show something about epsilo-delta technique but say's nothing about how using this technique, it look kind of random.

Thanks a lot Link-

PS. Sorry for my english, I am not a good english speaker or writter but at always I do my best :biggrin:
 
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  • #2
The easiest way to do this problem is to rewrite the function in polar coordinates. Then it should be easy to see that as r->0, the function goes to zero.
 
  • #3
StatusX said:
The easiest way to do this problem is to rewrite the function in polar coordinates. Then it should be easy to see that as r->0, the function goes to zero.

Thanks a lot.
 

1. What is the epsilon-delta technique used for in multi-variable limits?

The epsilon-delta technique is a mathematical method used to rigorously define and prove the limit of a function with multiple variables. It allows us to determine the behavior of a function as the input variables approach a certain point, by specifying a tolerance (epsilon) and finding a corresponding range (delta) of inputs that will ensure the function's output is within that tolerance.

2. How does the epsilon-delta technique differ from the single variable limit?

In the single variable limit, we only need to consider the behavior of a function as the input approaches a single value. However, in the multi-variable limit, we must take into account the behavior of the function as the inputs approach from any direction. This is why the epsilon-delta technique uses a tolerance and a range of inputs, rather than just a single value.

3. Can the epsilon-delta technique be used for all types of multi-variable limits?

Yes, the epsilon-delta technique can be applied to any type of multi-variable limit, including limits at a point, along a curve, or at infinity. It is a universal method for determining and proving the limit of a function regardless of the number of variables involved.

4. What is the role of epsilon and delta in the epsilon-delta technique?

Epsilon represents the tolerance or error bound that we want the function's output to be within. Delta represents the range of inputs that will ensure the function's output stays within that tolerance. By finding a suitable delta for a given epsilon, we can prove that the limit of the function exists at a certain point or along a certain path.

5. How is the epsilon-delta technique used in real-world applications?

The epsilon-delta technique is widely used in mathematical analysis and is essential for proving the existence and continuity of functions in calculus. It is also used in other fields such as physics, engineering, and economics to accurately determine the behavior of complex functions and make predictions based on the limits of those functions.

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