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The given problem presents the function:

\[
g(x) = \frac{x^2 - x - 6}{x^2 + 3x - 4}
\]

We are asked to evaluate the behavior of the limit as \( x \to -4^+ \) and determine which of the following is true:

1. \( \text{En } x = -4, \text{ el límite existe} \)  
2. \( x = -4 \text{ es una asíntota horizontal} \)  
3. \( El resultado del límite es un valor real \)  
4. \( x = -4 \text{ es una asíntota vertical} \)

### Step 1: Factorizing the numerator and denominator
We will first attempt to simplify the function by factoring both the numerator and denominator.

The numerator is \( x^2 - x - 6 \), which factors as:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

The denominator is \( x^2 + 3x - 4 \), which factors as:

\[
x^2 + 3x - 4 = (x - 1)(x + 4)
\]

So the function becomes:

\[
g(x) = \frac{(x - 3)(x + 2)}{(x - 1)(x + 4)}
\]

### Step 2: Analyze the limit \( \lim_{x \to -4^+} g(x) \)
The denominator has the term \( (x + 4) \), which causes the denominator to approach zero as \( x \to -4 \).

As \( x \to -4^+ \), the term \( (x + 4) \) becomes a very small positive number. Therefore, the overall function tends to infinity. Specifically, this suggests that the function exhibits a vertical asymptote at \( x = -4 \).

### Step 3: Conclusion
Since the function tends to infinity as \( x \to -4^+ \), this indicates that:

- The limit does not exist at \( x = -4 \).
- \( x = -4 \) is a vertical asymptote.

Thus, the correct answer is:

\[
\boxed{d. \ x = -4 \text{ es una asíntota vertical}}
\]

Would you like more details on any of these steps? Here are some related questions:

1. How do we identify vertical and horizontal asymptotes for rational functions?
2. What are the conditions for a limit to exist at a point of discontinuity?
3. Can the limit at \( x \to -4 \) be computed from both directions (left and right)?
4. How do the signs of the factors in the denominator affect the behavior of the function near the vertical asymptote?
5. What happens to the function if we evaluate the limit as \( x \to \infty \)?

**Tip**: Always factor both the numerator and denominator of a rational function before evaluating limits. It can reveal hidden simplifications and critical points like asymptotes.

Sea la función g(x) = (x^2 - x - 6) / (x^2 + 3x - 4). Al considerar lim (x → -4+) g(x), se verifica que:

Explore how to evaluate the limit lim (x → -4+) g(x) for the rational function g(x) = (x^2 - x - 6) / (x^2 + 3x - 4). This problem explains the identification of vertical asymptotes and the behavior of limits in rational functions, suitable for students in Grades 10-12.

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