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 such that f(x, y) is continuous at (1, -1), we need to evaluate the limit of f(x, y) as (x, y) approaches (1, -1) from all possible paths.

For function f(x, y) = sin(x + y)/(x^2 - y^2), let's evaluate the limit as (x, y) approaches (1, -1) from various paths:

1. Approach from a straight line path: Consider the path y = -x + 1. Substitute this equation into f(x, y) and simplify:
lim (x,y)→(1,-1) sin(x + y)/(x^2 - y^2)
= lim x→1 sin(x - x + 1)/((x^2 - (-x + 1)^2)
= lim x→1 sin(1)/((x^2 - (x^2 - 2x + 1))
= sin(1)/2

Since this limit is a constant value (sin(1)/2), the function f(x, y) is continuous at (1, -1) for any value of α.

For function f(x, y) = α, where (x, y) = (0, 0), we don't need to do any calculations. Since α is a constant value, the function f(x, y) is always equal to α at (0, 0). Thus, f(x, y) is continuous at (0, 0) for any value of α.

In conclusion, for both functions, any value of α will make them continuous at the respective points.