We are given the following limits:

$\lim_{{x \to a}} f(x) = 3 \quad \text{and} \quad \lim_{{x \to a}} g(x) = -4,$

and we are asked to evaluate the limit:

$\lim_{{x \to a}} \frac{-2[g(x)]^2}{f(x) + 5}.$

Step-by-step Solution:

1. Substitute the limit values of $f(x)$ and $g(x)$:

$f(x) \to 3 \quad \text{and} \quad g(x) \to -4 \quad \text{as} \quad x \to a.$

2. Evaluate the expression using these limit values:

[ \lim_{{x \to a}} \frac{-2[g(x)]^2}{f(x) + 5} = \frac{-2[-4]^2}{3 + 5}. ]

3. Simplify the expression:

• First, calculate $[-4]^2$: $[-4]^2 = 16.$

• Now, substitute back: $\frac{-2(16)}{3 + 5} = \frac{-32}{8}.$

4. Final simplification:

$\frac{-32}{8} = -4.$

Thus, the value of the limit is $-4$.

Answer:

The correct choice is A. $-4$.

Would you like more details or have any questions? Here are some related questions for further understanding:

1. What is the general procedure for evaluating limits involving fractions?
2. How do squared terms affect the outcome of limit problems?
3. What happens if one of the functions $f(x)$ or $g(x)$ approaches 0?
4. How would the limit change if the denominator approached 0?
5. Can limits of composite functions always be computed by substitution?

Tip: When solving limit problems, always check if direct substitution simplifies the expression. If not, consider algebraic simplification or limit laws.