The value of _ x 3 [ 1 x-3 + 9x 27-x^3 ] is:

Step-by-step solution

The given limit is _ x 3 [ 1 x-3 + 9x 27-x^3 ] Using the identity a^3 - b^3 = (a-b)(a^2+ab+b^2) , we can write 27 - x^3 = -(x^3 - 27) = -(x-3)(x^2+3x+9) Substituting this into the limit expression: _ x 3 [ 1 x-3 - 9x (x-3)(x^2+3x+9) ] Taking the common denominator: = _ x 3 [ x^2+3x+9-9x (x-3)(x^2+3x+9) ] = _ x 3 [ x^2-6x+9 (x-3)(x^2+3x+9) ] = _ x 3 [ (x-3)^2 (x-3)(x^2+3x+9) ] Since x 3 , x 3 , we can cancel the common factor (x-3) from the numerator and the denominator: = _ x 3 [ x-3 x^2+3x+9 ] Substituting x = 3 i