If f: R R is defined by f(x)= array cc 3 x- x x^2 , & for x 0 & , for x=0 array . and if f is continuous at x=0 , then is equal to

Step-by-step solution

Given that, f(x)= array cc 3 x- x x^2 , & for x 0 , & for x=0 array . Now, aligned LHL & = _ x 0⁻ f(x) & = _ x 0⁻ 3 x- x x^2 & = _ h 0 3(0-h)- (0-h) (0-h)^2 & = _ h 0 3 h- h h^2 & = _ h 0 -3 3 h+ h 2 h & = _ h 0 -9 3 h+ h 2 & = -9+1 2 =-4 aligned (using L' Hospital's rule) Since, f(x) is continuous at x=0 array ll & _ x 0⁻ f(x)=f(0) & -4= =-4 array