Final answer:
The existence of the limit of f(x, y) = x²y/(x² + y²) as (x, y) approaches (0, 0) would require consistency across all paths. Because the limit can differ along different paths, we conclude that the limit does not exist.
Step-by-step explanation:
The question asks whether the limit of the function f(x, y) = x²y/(x² + y²) exists as (x, y) approaches (0, 0). To determine this, one common method is to approach (0, 0) along different paths and see if the limit value is the same for each path. If the limit value changes depending on the path taken, then the overall limit does not exist.
For example, if we approach (0, 0) along the y-axis (x=0), the function simplifies to 0, since the numerator is 0 regardless of the value of y. However, if we approach along the line y=x, the function simplifies to x³/(2x²), which reduces to x/2, and as x approaches 0, this also approaches 0. Despite the same limit value along these two paths, we need to consider more paths to make a conclusion.
In general, if the limit of a function as it approaches a point is consistent across all paths, the limit exists. In this case, due to different limits along different paths to (0,0), the limit does not exist.