Let k be a non-zero real number. If f (x)= array cc ( e ^x-1 )^2 ( x k ) (1+ x 4 ) & , x 0 12 & , x=0 array . is a continuous function, then the value of k is

Step-by-step solution

f (x) is continuous at x=0 . aligned f (0) & = _ x 0 f (x) 12 & = _ x 0 ( e ^x-1 )^2 ( x k ) (1+ x 4 ) 12 & = _ x 0 ( e ^x-1 )^2 x^2 ( x k ) x (1+ x 4 ) x aligned aligned & 12= _ x 0 ( e ^x-1 x )^2 _ x 0 x k x k k _ x 0 (1+ x 4 )^ 1 x & 12= k _ x 0 ( e ^x-1 x )^2 _ x 0 x k x k _ x 0 _ x 0 [ (1+ x 4 )^ 4 x ]^ 1 4 aligned aligned & 12= k 1 ^ 1 4 [ array l _ x 0 e ^x-1 x =1 _ x 0 x x =1 _ x 0 (1+x)^ 1 x = e array ] & 12= k 1 4 & 12=4 k & k =3 aligned