Final answer:
To evaluate the limit of the function y³ / (y² + x²) as (x, y) approaches (0, 0), we need to analyze the behavior along different paths. If the limit as x approaches 0 is consistent across different paths, that is the limit's value; otherwise, the limit does not exist.
Step-by-step explanation:
The student is asking to evaluate the limit of the function f(x, y) = y³ / (y² + x²) as the point (x, y) approaches (0, 0). To analyze this limit, we should examine the behavior of the function as we approach the origin along different paths.
Step-by-step Evaluation
Consider approaching along the line y = mx, where m is a constant. Substituting this into our function gives:
f(x, mx) = (mx)³ / ((mx)² + x²) = m³x³ / (m²x² + x²) = m³x³ / x²(m² + 1) = m³x / (m² + 1).
As x → 0, it's clear that f(x, mx) → 0 regardless of the value of m.
We could also try other paths, like y = x² or y = µx² for some value µ, and check if the limit gives the same result or not.
If the limit varies depending on the path taken, the limit does not exist. If it remains constant, then that is the value of the limit.