Limits with three variables (a different problem)

In summary, the limit as (x,y)->0,0 of (x^2+y^2)ln(x^2+y^2) can be simplified to the limit of r->0 (from the right) of r^2 * (ln(r^2)), which evaluates to 0. This limit only needs to be evaluated from one direction (either from the right or the left) since r is always positive.
  • #1
physstudent1
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1

Homework Statement



the limit as (x,y)->0,0 of (x^2+y^2)ln(x^2+y^2)
(Hint: as (x,y)->(0,0) r->0(from the right)

Homework Equations





The Attempt at a Solution



I converted to polar coordinates then used trig identities and eventually got to the limit of r->0(from theright) of r^2 * (ln(r^2)) I eventually got this limit to equal 0. I'm pretty sure to make sure the limit exists I have to evaluate it as r->0 (from the left) as well but I'm not sure how...
 
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  • #2
You only need to evaluate the limit of r from one direction, as r is a variable that is always non-negative. Remember it represents a radius, ie a positive number.
 
  • #3
ohhhh I see thanks
 

FAQ: Limits with three variables (a different problem)

What is the definition of a limit with three variables?

A limit with three variables is a mathematical concept that describes the behavior of a function as it approaches a particular point in a three-dimensional space. It determines the value that a function is approaching as the three variables that define it approach specific values.

How do you evaluate a limit with three variables?

To evaluate a limit with three variables, you must first determine the approach direction of the three variables. Then, you can use algebraic manipulation and substitution to reduce the function to a two-variable limit. Finally, you can apply existing techniques for evaluating two-variable limits to find the value of the limit with three variables.

What are the common techniques for finding limits with three variables?

The most common techniques for finding limits with three variables include algebraic manipulation, substitution, and trigonometric identities. Other techniques such as L'Hopital's rule and the squeeze theorem may also be used in certain cases.

What are the key properties of limits with three variables?

The key properties of limits with three variables include the existence and uniqueness of the limit, continuity, and the limit laws. These properties dictate how a limit behaves and can be used to determine the value of a limit or prove its non-existence.

What are the real-world applications of limits with three variables?

Limits with three variables have various real-world applications in fields such as physics, engineering, and economics. They are used to model and analyze complex systems that involve multiple variables and determine the maximum or minimum values of a function in a three-dimensional space.

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