Evaluate `lim_(x to sqrt(2)) (x^(9) - 3x^(8) + x^(6) - 9x^(4) - 4x^(2) - 16x + 84)/(x^(5) - 3x^(4) - 4x + 12)`

Explanation

We have, `underset(x to sqrt(2))(lim) (x^(9) - 3x^(8) + x^(6) - 9x^(4) - 4x^(2) - 16 x + 84)/(x^(5) - 3x^(4) - 4x + 12)`
When `x = sqrt(2)`, the expression `(x^(9) - 3x^(8) + x^(6) - 9x^(4) - 4x^(2) - 16x + 84)/(x^(5) - 3x^(4) - 4x + 12)` takes the form `(0)/(0)`
`:. (x - sqrt(2))` is a factor of numerator and denominator. Since `(x - sqrt(2))` is an irratinal factor so its conjugate i.e., `x + sqrt(2)` will also be a factor of both numerator and denominator. Consequently. `x^(2) - 2` will be a common factor of numerator and denominator.
`:. underset(x to sqrt(2))(lim) (x^(9) - 3x^(8) + x^(6) - 9x^(4) - 4x^(2) - 16x + 84)/(x^(5) - 3x^(4) - 4x + 12)`
`= underset(x to sqrt(2)(lim) ((x^(2) - 2)(x^(7) - 3x^(6) + 2x^(5) - 5x^(4) + 4x^(3) - 19 x^(2) + 8x - 42))/((x^(2) - 2)(x^(3) - 3x^(2) + 2x - 6))`
`= underset(x to sqrt(2))(lim) (x^(7) - 3x^(6) + 2x^(5) - 5x^(4) + 4x^(3) - 19x^(2) + 8x - 42)/(x^(3) - 3x^(2) + 2x - 6)`
`= ((2sqrt(2)^(7) - 3(sqrt(2))^(6) + 2(sqrt(2))^(5) - 5 (sqrt(2))^(4) + 4 (sqrt(2))^(3) - 19 (sqrt(2))^(2) + 8 (sqrt(2)) - 42)/((sqrt(2))^(3) - 3 (sqrt(2))^(2) + 2 (sqrt(2))) - 6)`
`= (8 sqrt(2) - 24 + 8 sqrt(2) - 20 - 8 sqrt(2) - 38 + 8 sqrt(2) - 42)/(2sqrt(2) - 6 + 2 2 sqrt(2) - 6) = (32sqrt(2) - 124)/(4sqrt(2) - 12) = (8sqrt(2) - 31)/(sqrt(2) - 3)`