lim 108(x^2+2x)(x+1)^3
x-> -1 __________________ (divide)
(x^3+1)^3 (x-1)
To find the limit of the given expression as x approaches -1, we need to substitute -1 into the expression and simplify.
First, substitute -1 into the expression:
lim (108(x^2+2x)(x+1)^3) / ((x^3+1)^3 (x-1))
x-> -1
= 108*(-1^2+2*(-1))*(-1+1)^3 / ((-1^3+1)^3 (-1-1))
Now, simplify each component of the expression.
In the numerator:
= 108*(-1+2)*0^3 / ((-1+1)^3 (-2))
The term (-1+2)*0^3 in the numerator is equal to 0, because any term multiplied by 0 will be 0. And the term (-1+1)^3 in the denominator is also equal to 0, because any number subtracted from itself equals 0.
Therefore, the expression becomes:
= 0 / (0*(-2))
Since any number divided by 0 is undefined, the limit is undefined.
Hence, the limit of the given expression as x approaches -1 is undefined.