Final answer:
Using the Squeeze Theorem, we find the limit of the provided function as (x, y) approaches (9, 4) to be 0, since the term (x^2-81) approaches 0 and the cosine function is bounded between -1 and 1.
Step-by-step explanation:
To evaluate the limit using the Squeeze Theorem, we must find two functions that squeeze our given function. Since the cosine function oscillates between -1 and 1, we can say that:
-|x^2-81| ≤ (x^2-81)·cos(1/(x-9)^2 + (y-4)^2) ≤ |x^2-81|
As (x, y) approaches (9, 4), the term (x^2-81) approaches 0. So both the lower and upper bounding functions approach 0. Therefore, according to the Squeeze Theorem, the limit of the original function as (x, y) approaches (9, 4) is also 0.
It is not necessary to integrate or extensively use trigonometric identities here; we are simply using the properties of cosine and the bounds provided by its range to determine the limit of the entire function.