Limit of a two variable function

In summary, the conversation is about solving the limit of a function in R2, specifically lim (x,y)→(0,0) f(x,y) = lim [SUB](x,y)→(0,0) 3 * x2 / (x2 + y2). The person asking for help is unsure of how to proceed and suggests approaching the limit along different paths to prove that it does not exist. The other person responds by giving examples of approaching the limit along different paths and asking for the limits when x varies but y is constant and when y varies but x is constant.
  • #1
Jalo
120
0

Homework Statement


Hi.
I'm trying to solve the limit of the function:

lim (x,y)→(0,0)f(x,y) =
= lim (x,y)→(0,0) 3 * x2 / (x2 + y2)


Homework Equations





The Attempt at a Solution



x2 / (x2 + y2) is a limited function. Its image is always between 0 and 1. However I don't know what to do from here on.

What I tought was that since (x,y) → (0,0) then I can say that x = y and
x2 / (x2 + y2)
becomes 1/2. I don't think I can do this tho...

Any help or hint will be highly appreciated.
 
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  • #2
Unfortunately, because of the extra dimension, limits in R2 are much more difficult than in R. In R, you only have to prove that the limits "from above" and "from below" are equal. In R2, the limit as you approach the point on any path, even complicated curves, are the same.

Since you can't look at every possible curve, it is much easier to prove that a limit does NOT exist by showing that there exist different paths that give different limits. For example, IF you approach (0, 0) along the path y= x, you get [itex]x^2/(x^2+ y^2)= x^2/2x^2= 1/2[/itex] for all x so the limit is 1/2. Now, suppose you approach (0, 0) along the path (x, 0), the x-axis. What limit do you get? Although that should be enough, you could also look at the limit approaching (0, 0) along the line (0, y).
 
  • #3
What are the limits when you let x vary but hold y constant and when you let y vary but hold x constant?

*EDIT* Woops -- just saw that HallsofIvy beat me to the response.
 
  • #4
Nyah- Nyah!
 
  • #5
Thanks! That was very helpful!

D.
 

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

The limit of a two variable function is the value that a function approaches as the two input variables approach a specific point. It represents the behavior of the function near that point.

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

The limit of a two variable function can be calculated by approaching the specific point from different directions and observing the values of the function. If the values approach the same number from all directions, then that number is the limit of the function at that point.

3. Can a two variable function have more than one limit at a particular point?

Yes, a two variable function can have multiple limits at a specific point if the function approaches different values from different directions. In this case, the limit does not exist.

4. How is the limit of a two variable function affected by the behavior of the function at the specific point?

The behavior of the function at the specific point does not affect the limit of a two variable function. The limit is solely determined by the behavior of the function near that point.

5. What is the relationship between continuity and limits of a two variable function?

A two variable function is continuous at a point if the limit of the function at that point is equal to the value of the function at that point. Therefore, the existence of the limit at a point is a necessary condition for the function to be continuous at that point.

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