Example 1:

Here is a function of two variables that is not defined at zero:
(Note Abs[ ] is the function for absolute value

F[x_, y_] := Abs[x]/Sqrt[x^2 + y^2]

F[x, y]

Here mathematica computes the limit from the x-axis direction

Limit[F[x, 0], x -> 0]

and from the y-axis direction:

Limit[F[0, y], y -> 0]

They are different--- This means the limit does not exist. Here is the contour plot:

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}]

Here is another contour plot, As we approach from the x-direction we see different colors then as we approach from the y direction

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      ContourLines -> False, Contours -> 100, 
      PlotPoints -> 200]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, PlotPoints -> 40, 
      BoxRatios -> Automatic]

This plots the section of our surface lying above a strip along the x-axis

Plot3D[F[x, y], {x, -1, 1}, {y, - .1,  .1}, 
      PlotPoints -> {40, 7}, BoxRatios -> Automatic]

This plots the section of our surface lying above a strip along the y-axis

Plot3D[F[x, y], {x, - .1,  .1}, {y, -1, 1}, 
      PlotPoints -> {7, 40}, BoxRatios -> Automatic, 
      ViewPoint -> {0, -1, 0}]

Example 2:

Here is a function of two variables that is not defined at zero:

F[x_, y_] := (Abs[x] + Abs[y])/Sqrt[x^2 + y^2]

F[x, y]

Here mathematica computes the limit from the x-axis direction

Limit[F[x, 0], x -> 0]

and from the y-axis direction:

Limit[F[0, y], y -> 0]

But from the x=y line direction:

Limit[F[t, t], t -> 0]

They are different--- This means the limit does not exist. Here is the contour plot:

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}]

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      ContourLines -> False, Contours -> 100, 
      PlotPoints -> 100]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 71]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, PlotPoints -> 31, 
      ViewPoint -> {0, -1,  .4}]

Example 3

The following is an example where the function is not defined at (0,0) but the limit does exist

F[x_, y_] := x y/Sqrt[x^2 + y^2]

F[x, y]

Note: Mathematica won't take a square root of t^2!

F[t, t]

However you can tell it to:

PowerExpand[F[t, t]]

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}]

Example 4

The following is a composition of a quotient of polynomials with a continuous function. Therefore it is continuous wherever it is defined. In this case, it is defined whenever x≠y.

F[x_, y_] := Sin[(x^2 + y^2)/(x - y)]

F[x, y]

Here is a contour plot of this function:

ContourPlot[F[x, y], {x, -6, 6}, {y, -6.1, 6.1}]

Here is my attempt to pretty it up. Notice the contour plot "goes nuts" near the line x=y.

ContourPlot[F[x, y], {x, -6, 6}, {y, -6.1, 6.1}, 
      Contours -> 100, ContourLines -> False, 
      PlotPoints -> 200, ColorFunction -> Hue]

Here is the graph: 
(The errors come from Mathematica trying to plot where the function is not defined)

Plot3D[F[x, y], {x, -6, 6}, {y, -6, 6}, 
      PlotPoints -> 100]

Here we "zoom" in to the point (0,0)

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 100]

Here is a pretty picture of the first graph:

Plot3D[F[x, y], {x, -6, 6}, {y, -6, 6}, PlotPoints -> 500, 
      Mesh -> False]