The given function f(x,y) = (x + y^2) / (x^2 - y), has two independent variables. If the limit for the function f(x, y) approaching (0, 0) is defined, we should get the same value whether we take the values of (x, y) along x = 0, or (x, y) along y = 0.
If we follow x = 0
lim y--> 0 [ (y^2) / (- y) ] = -y
If we follow y = 0
lim x--> 0[ x/ x^2] = 1/x
This shows that the value of the function as (x,y) approach (0,0) is not the same.
The limit of the function at (0,0) does not exist.
To prove that the limit exists,we'll have to prove that the function approaches the same limit, even if the chosen path are different.
We'll approach (0,0) along x axis path => y = 0.
f(x,0) = x/x^2 = 1/x, for all x values that are different from 0.
We'll put now x = 0 and we'll approach along y axis path.
f(0,y) = y^2/-y =-y
Since the function has 2 different limits, along 2 different paths, therefore, the limit of the function f(x,y)=(x+y^2)/(x^2-y) does not exist.