Evaluate `lim_(x to 1) ((1)/(x^(2) + x - 2) - (x)/(x^(3) - 1))`

Explanation

When x = 1, the expression `(1)/(x^(2) + x - 2) - (x)/(x^(3) - 1)` takes the form
So, firstly we simply the expression to reduce it in the form `(0)/(0)`
`underset(x to 1)(lim) ((1)/(x^(2) + x - 2) - (x)/(x^(3) - 1)) underset(x to 1)(lim) ((1)/((x + 2)(x - 1)) - (x)/((x - 1)(x^(2) + x + 1)))`
`= underset(x to 1)(lim) (((x^(2) + x + 1)-x (x + 2))/((x + 2)(x - 1)(x^(2) + x + 1)))`
Now it is of the form `(0)/(0)`
`:. underset(x to 1)(lim) (((x^(2) + x + 1)-(x^(2) + 2x))/((x + 2)(x - 1)(x^(2) + x + 1))) = underset(x to 1)(lim) (-(x - 1))/((x + 2)(x - 1)(x^(2) + x + 1))`
`= underset(x to 1)(lim) (1)/((x + 2)(x^(2) + x + 1))`
`= (-1)/((1 + 2)(1 + 1+1))`
`(-1)/(9)`
`:. underset(x to 1)(lim) (-(1)/(x^(2) + x + 2) - (x)/(x^(3) - 1)) = - (1)/(9)`