Determining the limit for function of x and y

[WK95]

Homework Statement

For f(x,y) = (2x - y^2)/(2x^2 + y), what is the limit as (x,y)->(0,0)?

Homework Equations

The Attempt at a Solution

From this image, it seems that the limit would be non-existent since on one side of the sheet, it goes up and up to infinity whereas from the other side, it plunges down to negative infinity.

How can I show that the limit is DNE analytically?

 

[andrewkirk]

The general limit does not exist. However there are limits when one approaches (0,0) from certain directions, eg along the line y=x the limit is 2.

To prove the general limit does not exist, just pick a convenient direction, ie a line in the x-y plane, and then show that for any $\delta>0$, two points can be found on the line, both within distance $\epsilon$ from (0,0), for which the values of f(x,y) differ by more than 1. The line y=0 looks promising.