Limit of function of two variables

In summary, the problem is to find the limit of the function [x^2/(x^2 + y^2)] as (x,y) approaches (0,0). By plugging in y=0 and x=0, it can be seen that the limit does not exist. Using L'Hospital's Rule is not necessary in this case, as the limit can be determined by taking different paths and getting different answers. To confirm this, it is suggested to change to polar coordinates and see if the limit depends on the angle \theta.
  • #1
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Homework Statement


Fnd the limit, if it exists, or show that the limit does not exist.
lim [tex]_{(x,y)--> (0,0)} [x^2/(x^2 + y^2)] [/tex]


Homework Equations





The Attempt at a Solution



If x = 0, then f(0,y) = 0. f(x,y) --> 0 when (x,y) --> (0,0) along the y-axis.

If y = 0, then f(x,0) = 1. f(x,y) --> 1 when (,y) --> (0,0) along the x-axis.


1st, am I doing this right? By simply plugging in y = 0 or x = 0, I can determine the limit?

2nd, why am I not using L'Hospital's Rule. If (x,y) --> (0,0) then won't the function obviously go to 0/0 which is not real?
 
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  • #2
You are doing it right. An easier way to see what is going on to make the respective polar subsititutions and take the limit as r->0
 
  • #3
yes, you are correct. Indeed, the limit does not exist, since when taking different paths, the limits are different.
 
  • #4
Just to clarify: if you take the limit along two different paths and get different answers then the limit itself does not exist. If you take limits along many different paths and always get the same answer, that does not prove the limit exists because you can't try all possible paths. If you suspect the limit does exist, then the best thing to do is what end3r7 suggested: change to polar coordinates so that the distance from (0,0) is determined by the single variable r. If the limit, as r goes to 0, does not depend on [itex]\theta[/itex] the limit exists.
 
  • #5
Thanks .
 

1. What is the definition of a limit of a function of two variables?

The limit of a function of two variables is the value that the function approaches as the two input variables approach a specific point. This point is typically denoted by (a,b) and is known as the limit point.

2. How is the limit of a function of two variables calculated?

The limit of a function of two variables is calculated by evaluating the function at points that are close to the limit point (a,b). These points are then used to determine the behavior of the function as it approaches the limit point.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as it approaches the limit point from one side, either the left or the right. A two-sided limit, on the other hand, considers the behavior of the function from both sides simultaneously.

4. Can a limit of a function of two variables exist even if the function is not defined at the limit point?

Yes, a limit of a function of two variables can exist even if the function is not defined at the limit point. This is because the limit only concerns the behavior of the function as it approaches the limit point, not at the limit point itself.

5. How is the concept of a limit of a function of two variables applied in real-world situations?

In real-world situations, the concept of a limit of a function of two variables is used to model and predict the behavior of systems that involve two changing variables. For example, in physics, the position of an object can be described as a function of time and acceleration, and the limit of this function can be used to determine the object's final position.

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