Final answer:
The function f(x, y) is continuous except where xy = 0. To compute the limit as (x, y) approaches (2, 3), we substitute the values directly. The limit as (x, y) approaches (0, 0) does not exist due to the function's undefined behavior at that point.
Step-by-step explanation:
The function f(x, y) = x2 + y2 / xy should be analyzed for continuity. Firstly, the function is continuous everywhere except at points where xy = 0, since division by zero is undefined. Thus, the function is certainly not continuous at (x, y) = (0, 0). However, for any point where x and y are both non-zero, the function is continuous due to the continuity of the numerator and denominator separately.
(a) To compute the limit of f(x, y) as (x, y) approaches (2, 3), since the function is continuous at this point, we can simply substitute these values into the function:
f(2, 3) = 22 + 32 / (2 * 3) = 4 + 9 / 6 = 13 / 6
(b) When considering the limit of f(x, y) as (x, y) approaches (0, 0), since the function is not defined at this point, we must analyze the behavior of the function around this point. Different paths towards (0, 0) may yield different values, indicating that the limit does not exist. For instance, approaching along the x-axis (y=0) or the y-axis (x=0) will lead to a division by zero, which is undefined.