The value of _ x 0 ( 8 x^8 ) [1 - x^2 2 - x^2 4 + x^2 2 x^2 4 ] is equal to ...

Step-by-step solution

The given expression inside the limit can be factored as: (1 - x^2 2 ) (1 - x^2 4 ) The limit becomes: _ x 0 8 x^8 (1 - x^2 2 ) (1 - x^2 4 ) Using the standard limit _ 0 1 - ^2 = 1 2 , we can rewrite the expression as: _ x 0 8 x^8 ( 1 - x^2 2 ( x^2 2 )^2 x^4 4 ) ( 1 - x^2 4 ( x^2 4 )^2 x^4 16 ) Evaluating the standard limits: = _ x 0 8 x^8 ( 1 2 x^4 4 ) ( 1 2 x^4 16 ) = _ x 0 8 x^8 ( x^4 8 ) ( x^4 32 ) = _ x 0 8 x^8 x^8 256 = 8 256 = 1 32 Answer: 1 32