Final answer:
The limit of the function 4xy² / (3x²y²) as (x, y) approaches (0, 0) does not exist because the simplified form, 4 / (3x), approaches infinity as x approaches 0.
Step-by-step explanation:
The question involves evaluating the limit of a two-variable function as the point (x, y) approaches (0, 0). We are given the function 4xy² / (3x²y²). To find the limit as (x, y) approaches (0, 0), we can directly substitute x = 0 and y = 0 into the function. However, this substitution would result in a division by zero, which is undefined.
Instead, we notice that the y² terms in both the numerator and denominator can be canceled out, simplifying the expression to 4x / (3x²). We then further simplify by dividing both the numerator and denominator by x, which gives us 4 / (3x). Now, as x approaches 0, 4 / (3x) approaches infinity, and we can conclude that the function does not have a finite limit at the point (0, 0).
This behavior is analogous to functions that have asymptotes, where the function approaches infinity as the variable approaches a certain value.