Final answer:
The limit of the function as (x, y) approaches (0, 0) for the expression x⁸ - y⁸ / x⁴ + y⁴ is 0. This results from simplifying the expression by canceling out common terms and directly evaluating the simplified form.
Step-by-step explanation:
The question is asking for the limit of the function as (x, y) approaches (0, 0) of the expression x⁸ - y⁸ / x⁴ + y⁴. To find this limit, one can realize that the expression can be simplified by noting the numerator is a difference of two eighths powers, and the denominator is a sum of two fourths powers. Recognizing this, the following binomial factorization can be applied:
x⁸ - y⁸ = (x⁴ - y⁴)(x⁴ + y⁴)
Substituting this back into the original limit expression:
lim (x, y)→(0, 0) (x⁴ - y⁴)(x⁴ + y⁴) / (x⁴ + y⁴) = lim (x, y)→(0, 0) (x⁴ - y⁴)
Notice that the (x⁴ + y⁴) terms cancel each other out. This simplification leaves us with:
lim (x, y)→(0, 0) (x⁴ - y⁴)
Since both x⁴ and y⁴ approach 0 as x and y approach 0, the limit can now be directly evaluated as:
lim (x, y)→(0, 0) (x⁴ - y⁴) = 0⁴ - 0⁴ = 0
Therefore, the limit of the function as (x, y) approaches (0, 0) is 0.