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The image presents the following limit problem:

\[
\lim_{{x \to -4^+}} \frac{-4x}{x+4}
\]

### Solution:

Let's analyze the limit as \( x \to -4^+ \), which means \( x \) is approaching -4 from the right (values slightly greater than -4).

#### Step 1: Direct Substitution
If we directly substitute \( x = -4 \):

\[
\frac{-4(-4)}{-4 + 4} = \frac{16}{0}
\]

This results in division by zero, but because we're approaching from the right-hand side of -4, we need to check the sign of the result.

#### Step 2: Sign Analysis as \( x \to -4^+ \)
As \( x \) approaches -4 from the right, \( x + 4 \) becomes a small positive number (since values of \( x \) slightly greater than -4 give positive values for \( x + 4 \)).

Meanwhile, the numerator, \( -4x \), becomes negative because \( x \) is negative and multiplying by -4 yields a positive value. So, as \( x \to -4^+ \), the numerator stays positive while the denominator is a very small positive number.

#### Step 3: Conclusion
Since the numerator is positive and the denominator approaches 0 from the positive side, the overall expression approaches \( +\infty \).

\[
\lim_{{x \to -4^+}} \frac{-4x}{x+4} = +\infty
\]

### Final Answer:
The limit is \( +\infty \).

---

Would you like further details or have any other questions?

Here are 5 related questions you might want to explore:
1. How would the limit change if \( x \to -4^- \) (from the left)?
2. What if the function were \( \frac{-4x}{x - 4} \)?
3. Can we use L'Hôpital's Rule for limits like this?
4. How do one-sided limits affect the behavior of a function near discontinuities?
5. What is the significance of vertical asymptotes in rational functions like this?

**Tip**: When calculating limits involving division by zero, always consider the direction (left or right) from which \( x \) approaches the point, as this can determine whether the limit tends to \( +\infty \), \( -\infty \), or a finite value.

Learn how to calculate the one-sided limit of (-4x) / (x + 4) as x approaches -4 from the right. This problem involves analyzing division by zero, sign changes, and approaching infinity. Suitable for advanced high school students studying calculus.

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