The basic idea for limits of functions of one variable is that given
any ε we can find a δ such that if we are
within δ of x = a in the domain then f(x) lies between the lines y
= L  ε and y = L + ε . Formally  f(x)  L < ε whenever  xa  <
δ. But what about a function of 2 variables? I'll formalize
the language later but for now we'll consider the following 3 animations which
illustrate qualitatively what limits of functions of 2 variables are all about. The
first difference when we talk about functions of 2 variables is that instead of considering an interval containing x = a in
the domain of the form a  δ < x < a + δ We talk about a circle centered at a point (x_{0},y_{0}) of radius
δ in the domain of the form (xx_{0})^{2}
+ (yy_{0})^{2} < δ^{2} . Then
the definition of limit takes on a strikingly similar form: The limit of f(x,y) as (x,y)
approaches (x_{0},y_{0}) = L means that Given
any ε there is a δ such that the value of z = f(x,y) lies between the planes z = L  ε
and z = L + ε Whenever (x,y) is in the circle (xx_{0})^{2}
+ (yy_{0})^{2} < δ^{2} . See Animation
1  Notice as we shrink the circle (let δ go to 0) centered at (x_{0},y_{0})
the surface is confined between planes which are converging to z = 1.
Animation 1
Another way of thinking of limit is that if we approach
(x0,y0) along any curve in the domain we approach the same value
on the surface. In the following animation we consider the surface
z = 1  x^{2 } y^{2} (a parabaloid) , We consider what happens as we approach (0,0) along four different lines. Note on the surface all values converge to 1. See Animation 2
Animation 2
Of course this is not a proof we can't possibly
consider the limit of every curve which approaches a
point ( x0,y0). The formal proof goes like this: We want to show z =1 x^{2} y^{2} goes to 1 as (x,y)
goes to (0,0) Let ε be given then  f(x,y)  L  =  (1 
x^{2}  y^{2}) 1 = x^{2} + y^{2} Let
δ =√ ε then if x^{2 }+ y^{2} < δ^{2}
 f(x,y)  L  < ε We can though use this to prove a limit does not exist as the next animation shows.
If on the surface z approaches different values as we approach (x0,y0) along different paths then we say the limit does not
exist.See animation 3. You'll notice that all 3 lines approach (0,0) in the domain but on the surface the corresponding
z values approach 3 different values.
,
Animation 3
Why doesn't this limit exist ? Here f(x,y) = (xy)/(x^{2 }+ y^{2}) suppose we approach (0,0) along any line of the form y = m x f(x,mx)
= mx^{2}/[x^{2}(m^{2}+1)] = m/(m^{2}+1) hence the limit is m/(m^{2}+1)
which can take on any value in [1/2,1/2]
