If f (x)= 10^x+7^x-14^x-5^x 1- x , x 0 is continuous at x=0 , then the value of f(0) is

Step-by-step solution

Evaluating the limit as x 0 for continuity gives f(0) = _ x 0 10^x + 7^x - 14^x - 5^x 1 - x , an indeterminate form 0 0 . Factoring the numerator, 10^x + 7^x - 14^x - 5^x = (2^x - 1)(5^x - 7^x) . The limit simplifies using standard forms: _ x 0 a^x - 1 x = a and _ x 0 1 - x x^2 = 1 2 . Multiply and divide by x^2 : f(0) = _ x 0 (2^x - 1)(5^x - 7^x) x^2 x^2 1 - x First factor yields _ x 0 2^x - 1 x _ x 0 5^x - 7^x x = 2 ( 5 7 ) Second factor gives _ x 0 x^2 1 - x = 2 Combining results: f(0) = 2 2 ( 5 7 ) = 4 ( 5 7 )