Final answer:
L'Hospital's rule applies because the given limit results in an indeterminate form 0/0. After applying L'Hospital's rule, by taking the derivative of the numerator and denominator and evaluating at x = -2, the limit is determined to be -2.
Step-by-step explanation:
We must determine if L'Hospital's rule can be applied to the given limit:
lim (x → -2) [(x² + 8x + 12) / (x² + 2x)].
L'Hospital's rule is applicable when the limit leads to an indeterminate form such as 0/0 or ∞/∞. Let's evaluate the functions in the numerator and the denominator at the point x = -2.
The numerator: (-2)² + 8(-2) + 12 = 4 - 16 + 12 = 0,
The denominator: (-2)² + 2(-2) = 4 - 4 = 0.
Since both numerator and denominator evaluate to 0, the limit is indeed an indeterminate form 0/0, and we can apply L'Hospital's rule. By taking derivatives of the numerator and denominator and then evaluating at x = -2, we get:
The derivative of the numerator: 2x + 8,
The derivative of the denominator: 2x + 2.
Evaluating these at x = -2 gives 2(-2) + 8 = -4 + 8 = 4, and 2(-2) + 2 = -4 + 2 = -2, respectively. Thus, the limit is 4/-2 which simplifies to: -2