Final answer:
The limit of the function f(x, y) = xy/(x²+y²) as it approaches (0, 0) does not exist because the value of the limit depends on the path of approach.
Step-by-step explanation:
The question asks whether the limit of the function f(x, y) = xy/(x²+y²) exists as (x, y) approaches (0, 0). To determine if this limit exists, one common method is to approach (0, 0) along different paths and see if the limit is the same for all paths. For example, if we approach along the x-axis (y=0), the function simplifies to 0/x², which is 0. Approaching along the y-axis (x=0) gives us the same result, 0/y² = 0. However, if we try a different path, such as y=x, we get f(x, x) = x²/(2x²) = 1/2, which is clearly not 0. This indicates that the limit depends on the path of approach and hence the limit does not exist at the point (0, 0).