Limit Examples for Section 15.6

Below are the two examples from lecture where the limit of f(x,y) as (x,y) approaches (0,0) does not exist. The graphs on the left have x, y, z=f(x,y) between -5 and 5 for you to get an idea of the function’s behavior in general. The second graphs only show x>0 and y>0 with the paths x=αy with values α=1 (blue), α=2 (red), α=3 (green). In the first function, they approach different z values. In the second function, the z values grow without bound (tend to infinity) as (x,y) approaches (0,0).