Final answer:
The limit of the function (x² - 2xy + y²) / (x² + y²) as (x, y) approaches (0,0) does not exist, as demonstrated by evaluating the limit along two different paths and obtaining different limit values (0 and 1).
Step-by-step explanation:
The student is asking to compute the limit of the function (x² - 2xy + y²) / (x² + y²) as the point (x, y) approaches (0,0). To determine whether this limit exists, we can investigate the behavior of the function along different paths towards the origin. If the limit values along these paths differ, the overall limit does not exist.
Firstly, consider the path where x = y. Substituting y for x, we get:
((y² - 2y² + y²) / (y² + y²)) = (0 / (2y²)) = 0
As y approaches 0, the limit along this path is 0. Now, consider the path where y = 0. Substituting 0 for y, we get:
((x² - 2x(0) + (0)²) / (x² + (0)²)) = (x² / x²) = 1
As x approaches 0, the limit along this path is 1. Since the limits along these two paths are different (0 and 1), the overall limit does not exist.