determine the value of the constant α so that f(x,y) is continuous at (1,-1) if
f(x,y):
i)sin(x+y)/(x^(2 )– y^2 (x,y)≠(0,0)
ii) α (x,y)=(0,0)
To determine the value of the constant α so that f(x, y) is continuous at (1, -1), we need to ensure that the function f(x, y) approaches the same value from all directions as (x, y) approaches (1, -1).
i) For the function f(x, y) = sin(x + y) / (x^2 - y^2), we need to check the limit of the function as (x, y) approaches (1, -1).
To find the limit, we can approach (1, -1) along different paths. Let's consider approaching from two different directions: along the x-axis (y = -1) and along the y-axis (x = 1).
Approaching along the x-axis (y = -1):
lim(x->1) f(x, -1) = lim(x->1) sin(x + (-1)) / (x^2 - (-1)^2)
= sin(1 + (-1)) / (1^2 - (-1)^2)
= sin(0) / (1 - 1)
= 0 / 0 (indeterminate form)
Approaching along the y-axis (x = 1):
lim(y->-1) f(1, y) = lim(y->-1) sin(1 + y) / (1^2 - y^2)
= sin(1 + (-1)) / (1 - (-1)^2)
= sin(0) / (1 - 1)
= 0 / 0 (indeterminate form)
From our limits, we can see that the function is undefined or indeterminate at (1, -1) for any value of x and y ≠ (0, 0). Therefore, there is no value of α that can make this function continuous at (1, -1).
ii) For the function f(x, y) = α, when (x, y) = (0, 0), we need to check if it approaches a constant value as (x, y) approaches (0, 0).
Since the function is simply a constant α, it is already continuous at (0, 0) for any value of α because it does not depend on the values of x and y. Therefore, any value of α will make this function continuous at (0, 0).