We have to find lim x--> -2 [(x^2-2x-8)/(x^3+8)] using l'Hopital's rule.

First we find out if the l'Hopital's Rule can be used here. Substituting x = -2 we get the indeterminate form 0/0; therefore it can. We now use the derivatives of the numerator and the denominator.

lim x--> -2 [ (2x - 2)/(3x^2)]

substitute x = -2

=> (-4 - 2)/12

=> -6/12

=> -1/2

**The required value is lim x--> -2 [(x^2-2x-8)/(x^3+8)] = -1/2**

We know that l'Hospital theorem could be applied if the limit gives an indetermination.

We'll verify if the limit exists, for x = -2.

We'll substitute x by -2 in the expression of the function.

lim y = lim (x^2-2x-8)/(x^3+8)

lim (x^2-2x-8)/(x^3+8) = (4+4-8)/(-8+8) = 0/0

We've get an indetermination case.

We'll apply L'Hospital rule:

lim f(x)/g(x) = lim f'(x)/g'(x)

f(x) = x^2-2x-8 => f'(x) = 2x-2

g(x) = x^3+8 => g'(x) = 3x^2

lim (x^2-2x-8)/(x^3+8) = lim (2x-2)/3x^2

We'll substitute x by -2:

lim (2x-2)/3x^2 = (-4-2)/12

lim (2x-2)/3x^2 = -6/12

lim (2x-2)/3x^2 = -1/2

**The limit of the function, for x->-2, is: lim (x^2-2x-8)/(x^3+8) = -1/2.**