What is the limit of (x cubed - 8) divided by (4 - x squared) as x approaches 2?

To find the limit of the expression ((x^3 - 8) / (4 - x^2)) as (x) approaches 2, we first substitute (x = 2) directly into the expression.

Calculating the numerator:

[

x^3 - 8 = 2^3 - 8 = 8 - 8 = 0

]

Now for the denominator:

[

4 - x^2 = 4 - 2^2 = 4 - 4 = 0

]

This results in the indeterminate form (0/0), which indicates that further analysis is necessary. To resolve this, we should factor both the numerator and the denominator.

The numerator can be factored using the difference of cubes formula, (a^3 - b^3 = (a - b)(a^2 + ab + b^2)), where (a = x) and (b = 2):

[

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

]

Now, let's factor the denominator. The expression (4 - x^2) can be