f (x)= cases [x^2 ]- [-x^2 ], & x 3 k & , x=3 cases is continuous at x=3 , then k = where [ ] is greatest integer function

Step-by-step solution

A function is continuous at a point if the limit exists and equals the function value there. For f(x) defined as f(x) = array ll [x^2 ] - [-x^2 ], & x 3 k, & x = 3 array . , we have f(3) = k . To determine continuity at x=3 , consider the left-hand limit. As x 3^- , x^2 9^- , so [x^2 ] = 8 and [-x^2 ] = -9 . Thus, _ x 3^- f(x) = 8 - (-9) = 17 . For the right-hand limit, as x 3^+ , x^2 9^+ , so [x^2 ] = 9 and [-x^2 ] = -10 . Therefore, _ x 3^+ f(x) = 9 - (-10) = 19 . Since the left-hand and right-hand limits are une