Proving that a limit of a two-variable function does not exist

In summary, the conversation discusses finding the limit of a function as (x,y) approaches (0,0) and the difficulty in proving that the limit does not exist. It also poses a challenge to find a value of b for which the function on the line y=bx blows up at all nonzero values of x.
  • #1
bellerevolte
1
0
Find the limit as (x,y) -> (0,0) of (x^4 + y^4)/(x^3 + y^3)

This was a question from a recent homework set (class homework is done online), and the server accepted 0 as an answer. However, the actual answer is that the limit does not exist. My professor told us this afterwards and proposed that we find a way to prove that the limit indeed does not exist (I'm assuming this means to find a function from which the limit does not approach 0). But every function I have tried so far ends up making the limit 0.

Anyone up for a challenge? :)
 
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  • #2
Consider the line y=bx. Can you find a value of b such that the function blows up at all nonzero values of x on this line?
 

1. What does it mean for a limit of a two-variable function to not exist?

When a limit of a two-variable function does not exist, it means that as the two input variables in the function approach a certain point, the output values do not approach a single constant value. In other words, the function behaves unpredictably at that point.

2. How can you prove that a limit of a two-variable function does not exist?

To prove that a limit of a two-variable function does not exist, you can use a variety of techniques such as the epsilon-delta definition, the sequential criterion, or the polar coordinate criterion. These methods involve finding a specific point or sequence of points where the function does not approach a single limit.

3. Can a two-variable function have a limit in one direction but not in another?

Yes, a two-variable function can have a limit in one direction but not in another. This is known as a one-sided limit. In this case, the function approaches a specific value as the two variables approach from one side, but not from the other side.

4. Are there any common types of two-variable functions that do not have limits?

Yes, there are some common types of two-variable functions that do not have limits. These include functions with a discontinuity or a jump, functions with an infinite or oscillatory behavior, and functions with a corner or cusp. These types of functions may not have a limit at certain points.

5. What is the significance of proving that a limit of a two-variable function does not exist?

Proving that a limit of a two-variable function does not exist is important because it helps to understand the behavior of the function at certain points. It also allows us to identify and analyze any discontinuities or irregularities in the function. Additionally, it is a fundamental concept in calculus and is necessary for solving more advanced problems involving limits and continuity.

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