The given problem asks you to find the limit:

$\lim_{x \to -1^+} \frac{(x+5) \cdot \frac{|x+1|}{x+1}}{}$

Since we are taking the limit as $x$ approaches $-1$ from the right ($x \to -1^+$), let's break it down:

Step 1: Simplify the absolute value term

• The absolute value function $|x + 1|$ behaves differently depending on whether $x + 1$ is positive or negative.
• Since $x \to -1^+$ means $x$ is slightly greater than $-1$, $x + 1$ is positive for $x \to -1^+$. Thus, $|x + 1| = x + 1$.

Step 2: Substitute into the expression

So the given expression simplifies to:

$\frac{(x + 5) \cdot \frac{x + 1}{x + 1}}{}$

For $x \neq -1$, the $\frac{x + 1}{x + 1}$ cancels out, leaving:

$x + 5$

Step 3: Take the limit

Now we take the limit of $x + 5$ as $x \to -1^+$:

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

Thus, the limit is $4$.

Conclusion:

The correct answer is C: 4.

Would you like any more details or have any questions? Here are some related questions to consider:

1. How does the behavior of the absolute value function change if the limit is from the left?
2. What happens if the limit was as $x$ approaches $-1$ from both sides (two-sided limit)?
3. Can this type of problem have an undefined limit?
4. What is the general strategy for handling limits involving absolute values?
5. What are the differences between one-sided and two-sided limits?

Tip: Always carefully analyze the direction of limits when absolute values are involved!