If f (x) is continuous at point x=0 where f (x)= array cl 3 x+5 x a^x-1 & , x 0 array . then the values of a and b , respectively, are

Step-by-step solution

The function f(x) is continuous at x=0 when _ x 0^- f(x) = _ x 0^+ f(x) = f(0) = 2 2 , with interpreted as the natural logarithm. Evaluating the left-hand limit: _ x 0^- 3 x + 5 x a^x - 1 = _ x 0^- 3 x x + 5 x x a^x - 1 x Using standard limits _ x 0 x x = 1 , _ x 0 x x = 1 , and _ x 0 a^x - 1 x = a , we obtain 8 a . For the right-hand limit: _ x 0^+ 8x + 2x x b^x - 1 = _ x 0^+ 8 + 2 x b^x - 1 x Applying _ x 0 x = 1 and _ x 0 b^x - 1 x = b gives 10 b . Equating both limits to f(0) : 8 a = 2 2 8 2 = 2 a a = 4 2 a = 1