Final answer:
To find the limit as (x, y) approaches (0, 0) for the function f(x, y) = xy / (x² + y²), we consider approaching (0, 0) along various paths to determine if the limit is the same. The limit of the function along the x-axis and y-axis is 0, but we cannot conclude that the limit exists without analyzing all possible paths or applying specific limit theorems.
Step-by-step explanation:
The question asks to find the limit of the function f(x, y) = xy / (x² + y²) as (x, y) approaches (0, 0). To evaluate this limit, one must consider different paths approaching (0, 0) and see if the limit is the same along all paths. If the limit varies depending on the path taken, the limit does not exist.
Let's take two different paths to approach (0, 0): For example, approach along the x-axis (y = 0) and along the y-axis (x = 0).
- Path 1: Along the x-axis (y = 0), the function reduces to 0 since the numerator becomes 0.
- Path 2: Along the y-axis (x = 0), the function also reduces to 0 since the numerator is 0.
However, to confirm that the limit exists, we need to show that the limit is the same for any path to (0, 0). This can be more complex and requires additional analysis that cannot be concluded from just two paths.
In many cases, to prove the existence of a limit, one may use polar coordinates or apply the definition of the limit directly. For more definitive results, a detailed analysis or application of specific mathematical theorems might be necessary.