_ x 1 [ x-2 x^2-x - 1 x^3 - 3x^2 + 2x ] =

Step-by-step solution

The given limit is _ x 1 [ x-2 x^2-x - 1 x^3 - 3x^2 + 2x ] Factorizing the denominators, we get: = _ x 1 [ x-2 x(x-1) - 1 x(x-1)(x-2) ] Taking the common denominator x(x-1)(x-2) : = _ x 1 [ (x-2)^2 - 1 x(x-1)(x-2) ] Expanding the numerator: = _ x 1 [ x^2 - 4x + 4 - 1 x(x-1)(x-2) ] = _ x 1 [ x^2 - 4x + 3 x(x-1)(x-2) ] Factorizing the numerator: = _ x 1 [ (x-1)(x-3) x(x-1)(x-2) ] Canceling the common factor (x-1) since x 1 implies x 1 : = _ x 1 x-3 x(x-2) Substituting x = 1 into the simplified expression: = 1-3 1(1-2