Limit of a two variable function

[Jalo]

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.

 

[HallsofIvy]

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 $x^2/(x^2+ y^2)= x^2/2x^2= 1/2$ 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).

 

[jmcelve]

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.

 

[HallsofIvy]

Nyah- Nyah!

 

[Jalo]

Thanks! That was very helpful!

D.