_ x 0 63^x-9^x-7^x+1 2 - 1+ x = .

Step-by-step solution

To evaluate _ x 0 63^x - 9^x - 7^x + 1 2 - 1 + x , we first observe that substituting x = 0 gives the indeterminate form 0 0 as both numerator and denominator vanish. The numerator factors as (9^x - 1)(7^x - 1) by writing 63^x as 9^x 7^x and grouping terms. Rationalizing the denominator gives 1 - x 2 + 1 + x , which using the identity 1 - x = 2 ^2 ( x 2 ) becomes 2 ^2 ( x 2 ) 2 + 1 + x . Thus, the limit becomes _ x 0 (9^x - 1)(7^x - 1)( 2 + 1 + x ) 2 ^2 ( x 2 ) . Rewriting for standard limits: _ x 0 9^x - 1 x 7^x -