Final answer:
To find the limit of the given function, we can use the technique of polar coordinates. By substituting x = r cosθ and y = r sinθ and letting (r, θ) approach (0, 0), we can simplify and evaluate the expression to find that the limit is -4.
Step-by-step explanation:
To find the limit of the given function, we can use the technique of polar coordinates. We substitute x = r cosθ and y = r sinθ, and let (r, θ) approach (0, 0) as (x, y) approaches (0, 0). The limit expression becomes:
lim (r, θ)→(0, 0) cos²θ sin²θ / (cos²θ + sin²θ)¹⁶ - 4
Since cos²θ + sin²θ = 1, the expression becomes:
lim (r, θ)→(0, 0) cos²θ sin²θ / (1)¹⁶ - 4
Simplifying further, we get:
lim (r, θ)→(0, 0) cos²θ sin²θ / 1 - 4
Next, we can use the double angle identities for sine and cosine to rewrite the expression. The double angle identities state that sin²θ = (1 - cos(2θ))/2 and cos²θ = (1 + cos(2θ))/2. Substituting these identities into the limit expression:
lim (r, θ)→(0, 0) ((1 + cos(2θ))/2) ((1 - cos(2θ))/2) / 1 - 4
Expanding and simplifying the expression gives:
lim (r, θ)→(0, 0) (1 - cos²(2θ))/4 - 4
Since 1 - cos²(2θ) = sin²(2θ), the expression becomes:
lim (r, θ)→(0, 0) sin²(2θ)/4 - 4
At this point, we can substitute back the original variables x and y to get:
lim (x, y)→(0, 0) sin²(2θ)/4 - 4
Finally, as (x, y) approaches (0, 0), θ approaches 0 as well. Therefore, we can substitute 0 for θ in the expression:
lim (x, y)→(0, 0) sin²(2(0))/4 - 4
Substituting θ = 0 gives:
lim (x, y)→(0, 0) sin²(0)/4 - 4
Simplifying sin²(0) gives:
lim (x, y)→(0, 0) 0/4 - 4
And further simplification gives:
lim (x, y)→(0, 0) -4
Therefore, the limit of the given expression as (x, y) approaches (0, 0) is -4.