Final answer:
The limit of the function f(x, y) = (x⁴ - 12y²) / (x² + 6y²) as (x, y) approaches (0, 0) does not exist because it approaches different values along the x-axis and the y-axis.
Step-by-step explanation:
The student has asked to find the limit of the function f(x, y) = (x⁴ - 12y²) / (x² + 6y²) as (x, y) approaches (0, 0). To determine if this limit exists, one method is to approach the origin along different paths and see if the limit is the same along each path.
Limit Along the x-axis (y=0)
When y = 0, the function simplifies to f(x, 0) = x⁴ / x² = x², which approaches 0 as x approaches 0.
Limit Along the y-axis (x=0)
When x = 0, the function becomes f(0, y) = -12y² / 6y² = -2, which is -2 for any y ≠ 0.
Since the limit along the x-axis is different from the limit along the y-axis, the overall limit of f(x, y) as (x, y) approaches (0, 0) does not exist.