Proving that a limit is non-existent

[Mathoholic!]

Homework Statement

The function is:
f: D={(x,y)\inℝ²:x+y≠0}→ℝ
(x,y)→\frac{x-y}{x+y}

They ask you to prove that the limit as (x,y)→(0,0) is non-existent.

Homework Equations

The Attempt at a Solution

My attempt at a solution was using the definition of limit: If there was a limit (L\inℝ) when (x,y)→(0,0), then \forallε>0 there would be a δ>0 for which:

||(x,y)||<δ →(implied) |f(x,y)-L|<ε

I tried guessing an ε for which there was no δ, hence proving the non-existence of the limit, but I can't seem to find it.

Another attempt at a solution was changing the expression to polar coordinates which gave me the following:

f(r,θ)=\frac{cosθ-senθ}{cosθ+senθ} \forallr>0

The limit when r→0 is always the same: f(r,θ). But since the function oscillates indeterminately, there's no limit.

Am I doing this right? I really need some good feedback on this ;)

 

[Joffan]

You have some interesting ideas. My simple version would be to note that if you approach (0,0) along the two axes, you get different limits.

 

[HallsofIvy]

It should be sufficient to point out that there exist points (x, y), on the line y= x, arbitrarily close to (0, 0) such that f(x, y)= 0 and, on the line y= -x, arbtririly close to (0, 0), for which f(x, y) is not defined.