Final answer:
The limit of the function f(x, y) = (x⁴ - 34y²) / (x² + 17y²) as (x, y) approaches (0, 0) exists and is equal to 1.
Step-by-step explanation:
To find the limit of the function f(x, y) = (x⁴ - 34y²) / (x² + 17y²) as (x, y) approaches (0, 0), we look for ways to simplify the expression or determine its behavior by substituting values that approach zero. By plugging in x = 0 and y = 0 directly, we get an indeterminate form 0/0, which suggests that we need to analyze the limit further. However, we can observe that the highest powers of x and y in the numerator and denominator are the same, which indicates the limit could be simplified.
Substituting y = mx where m is an arbitrary constant, the function becomes (x⁴ - 34(m²)x²) / (x² + 17(m²)x²) = (x²(1 - 34m²)) / (x²(1 + 17m²)). After simplifying, we can cancel out x², which is nonzero for values of x not equal to zero, and we are left with (1 - 34m²) / (1 + 17m²). Since m is constant as x approaches zero, the limit of the function as (x, y) approaches (0, 0) is essentially the limit of (1 - 34m²) / (1 + 17m²) as x approaches zero, which evaluates to 1 since m² and, consequently, 34m² and 17m² are approaching zero.
Thus, the limit exists and is equal to 1 regardless of the path taken as (x, y) approaches (0, 0).