If f (x)= ( ^2 x ) 3 x^2 , x 0 is continuous at x=0 then f (0)=

Step-by-step solution

To ensure continuity at x = 0 , we require f (0) = _ x 0 f (x) , where f (x) = ( ^2 x ) 3 x^2 for x 0 . Rewriting using ^2 x = 1 - ^2 x , the argument becomes ^2 x = (1 - ^2 x) = - ^2 x . Using the identity ( - ) = , we have ( ^2 x) = ( ^2 x) . The limit evaluates as: _ x 0 ( ^2 x) 3 x^2 = _ x 0 [ ( ^2 x) ^2 x ^2 x 3 x^2 ] Applying standard limits : as x 0 , ^2 x 0 , so the first factor tends to 1. The second factor simplifies to 3 ( x x )^2 3 . Therefore, f (0) = 3 . 3