Let [t] denote the greatest integer less than or equal to t . A function f(x) is defined as: f(x) = array cl a^2 [ (1+x) x ] + b [ 1-e^x x ], & x 0 array . If f(x) is continuous at x=0 and a is an integer, then the value of a+b is

Step-by-step solution

For continuity at x=0 , _ x 0^- f(x) = _ x 0^+ f(x) = f(0) = 14 . Evaluating the Left Hand Limit (LHL): Using Taylor series expansions for x 0^- : (1+x) x = x - x^2 2 + x^3 3 - x = 1 - x 2 + x^2 3 - Since x 0 , so (1+x) x approaches 1 from a value slightly greater than 1 . Thus, [ (1+x) x ] = 1 . Similarly, 1-e^x x = 1 - (1 + x + x^2 2 + ) x = -1 - x 2 - Since x 0 , so 1-e^x x approaches -1 from a value slightly greater than -1 . Thus, [ 1-e^x x ] = -1 . Therefore, LHL = a^2(1) + b(-1) = a^2 - b . Equating to f(0)