If f (x)= array cc 9^x-2 3^x+1 (1+3 x) 2 x & , if x 0 a ( b )^ c & , if x=0 array . is continuous at x=0 , then a + b + c =

Step-by-step solution

To ensure continuity at x = 0 , the limit must satisfy _ x 0 f(x) = f(0) . Direct evaluation gives f(0) = a ( b)^c . The limit expression is _ x 0 9^x - 2 3^x + 1 (1 + 3x) 2x . Factor the numerator: 9^x - 2 3^x + 1 = (3^x - 1)^2 . Rewriting the limit: _ x 0 (3^x - 1)^2 (1 + 3x) 2x . Using known limits _ x 0 a^x - 1 x = a , _ x 0 (1 + kx) x = k , and _ x 0 kx x = k , divide by x^2 : _ x 0 ( 3^x - 1 x )^2 ( (1 + 3x) x ) ( 2x x ) = ( 3)^2 3 2 = ( 3)^2 6 . Setting f(0) = _ x 0 f(x) yields a ( b)^c = ( 3)^2 6 . Matching