We are tasked with finding the limit:

$\lim_{x \to -1} \frac{x^2 + 5x - 4}{x^2 - 3x - 4}$

Step 1: Direct Substitution

First, substitute $x = -1$ into the expression:

$\frac{(-1)^2 + 5(-1) - 4}{(-1)^2 - 3(-1) - 4} = \frac{1 - 5 - 4}{1 + 3 - 4} = \frac{-8}{0}$

The denominator equals zero, which suggests that we may have a vertical asymptote or that the function approaches infinity. Let’s attempt to factor both the numerator and the denominator to see if we can simplify the expression.

Step 2: Factor the expression

Factor the numerator and denominator.

Numerator:

The quadratic expression in the numerator is $x^2 + 5x - 4$. This does not factor easily, so we leave it as is for now.

Denominator:

The quadratic expression in the denominator is $x^2 - 3x - 4$, which factors as:

$x^2 - 3x - 4 = (x - 4)(x + 1)$

Step 3: Simplify the expression

Now, we have the limit:

$\lim_{x \to -1} \frac{x^2 + 5x - 4}{(x - 4)(x + 1)}$

As $x \to -1$, we see that the denominator $(x + 1)$ approaches zero. This confirms that the function has a vertical asymptote at $x = -1$, and the limit does not exist in the finite sense. However, we can determine if the function tends toward $+\infty$ or $-\infty$ by examining the behavior around $x = -1$.

Step 4: Analyze the sign near $x = -1$

• As $x$ approaches $-1$ from the left ($x \to -1^-$):

  • The term $(x + 1)$ is negative (since $x + 1 < 0$),
  • The numerator $x^2 + 5x - 4$ is positive (since the parabola opens upwards and does not cross zero near $x = -1$).

  Therefore, the overall expression tends to $-\infty$.

• As $x$ approaches $-1$ from the right ($x \to -1^+$):

  • The term $(x + 1)$ is positive (since $x + 1 > 0$),
  • The numerator $x^2 + 5x - 4$ remains positive.

  Therefore, the overall expression tends to $+\infty$.

Conclusion

The limit does not exist because the function tends to $-\infty$ from the left and $+\infty$ from the right.

Would you like more details on any specific steps?

Here are some related questions to explore:

1. How do you determine limits involving asymptotes?
2. What is the formal definition of a limit not existing?
3. Can limits involving rational functions always be simplified?
4. How does factoring help in limit problems?
5. What is the significance of vertical asymptotes in limit problems?

Tip: Always try direct substitution first to identify indeterminate forms or zero denominators, then proceed with factoring or algebraic simplifications.