Difficult Multivariate limit problem

In summary, the task is to determine if the limit of yln(x^2 + y^2) as (x,y) approaches (0,0) exists or not. The attempt at a solution involves using L'Hopital's rule, but it seems that the y factor decreases faster than the ln(x^2 + y^2) factor. The use of polar coordinates reveals that the limit will be equal to 0 if and only if the limit of rln(r) as r approaches 0 is also 0.
  • #1
Scriabin11
1
0

Homework Statement



I am supposed to show whether the limit:

lim (x,y)----> (0,0) of yln(x^2 + y^2) exists or doesn't.

Homework Equations





The Attempt at a Solution




I've tried numerous paths, but what it seems to come down to is showing that the y factor goes to zero quicker than the ln(x^2 +y^2) factor. And I don't believe L'Hopitals rule is applicable.
 
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  • #2
L'hospitals rule is applicable since you end of with inderterminate forms when you approach x and y seperately.
 
  • #3
In polar coordinates, [itex]y ln(x^2+ y^2)[/itex] becomes [itex]r sin(\theta) ln(r^2)= (2 r ln(r))sin(\theta)[/itex]. That will go to 0 independently of y if and only if [itex]\lim_{r\to 0} r ln(r)= 0[/itex]. Is that true?
 

Related to Difficult Multivariate limit problem

1. What is a multivariate limit problem?

A multivariate limit problem is a mathematical concept that involves finding the limit of a function as multiple variables approach a specific value. In simpler terms, it is the value that a function approaches as the input variables get closer and closer to a certain point.

2. How is a multivariate limit problem different from a regular limit problem?

A multivariate limit problem is different from a regular limit problem because it involves multiple variables instead of just one. This adds an extra level of complexity as each variable needs to approach the specific value simultaneously in order to determine the limit.

3. What are some common techniques for solving difficult multivariate limit problems?

Some common techniques for solving difficult multivariate limit problems include using L'Hôpital's rule, factoring and simplifying the expression, and using substitution. It is important to use a combination of these techniques and select the most appropriate one for each specific problem.

4. What are some real-world applications of multivariate limit problems?

Multivariate limit problems have many real-world applications, particularly in the fields of physics, engineering, and economics. They are used to model and predict complex systems such as the weather, traffic patterns, and stock market trends.

5. How can I improve my understanding of multivariate limit problems?

To improve your understanding of multivariate limit problems, it is important to practice solving various types of problems and to seek help from a tutor or teacher if needed. Additionally, studying the underlying concepts and principles of multivariate calculus can also help in better understanding these types of problems.

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