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The problem asks to find the limit of the function \( \frac{-5x}{x + 6} \) as \( x \) approaches \(-6^+\) (from the right side).

### Step-by-Step Solution:

1. **Understand the Direction of the Limit**:
   - The notation \( x \to -6^+ \) indicates that \( x \) approaches \(-6\) from values that are slightly greater than \(-6\) (i.e., from the right side of \(-6\)).

2. **Substitute \( x = -6 \)**:
   - First, let's directly substitute \( x = -6 \) into the expression:
     \[
     \frac{-5(-6)}{-6 + 6} = \frac{30}{0}
     \]
   - This results in a division by zero, but the sign of the result depends on the direction from which \( x \) approaches \(-6\).

3. **Analyze the Sign of the Expression**:
   - As \( x \) approaches \(-6\) from the right (i.e., \( x > -6 \)), \( x + 6 \) is a small positive number. So:
     \[
     \frac{-5x}{x + 6}
     \]
     - The numerator \( -5x \) will be negative because \( -5x \) is negative when \( x \) is positive.
     - The denominator \( x + 6 \) is a small positive number.

4. **Conclusion**:
   - When \( x \to -6^+ \), \( \frac{-5x}{x + 6} \) tends towards \( -\infty \) because the numerator is negative, and the denominator is a small positive number.

### Final Answer:
\[
\lim_{x \to -6^+} \frac{-5x}{x + 6} = -\infty
\]

Would you like more details or have any questions?

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Here are five related questions that expand on this topic:

1. How would the limit change if \( x \) approached \(-6\) from the left side, i.e., \( x \to -6^- \)?
2. What is the difference between limits approaching from the left, right, and both sides?
3. How do you determine the behavior of a limit that leads to infinity?
4. Can you explain how to solve a limit that results in an indeterminate form like \( \frac{0}{0} \)?
5. What happens to the limit if the numerator and denominator have opposite signs as \( x \) approaches the limit point?

**Tip:** When evaluating limits that result in infinity, always consider the direction from which the variable is approaching to determine the sign and nature of the limit (positive or negative infinity).

Find the limit: \( \lim_{x \to -6^+} \frac{-5x}{x + 6} \).

Explore the step-by-step solution to find the limit of \( \frac{-5x}{x + 6} \) as x approaches -6 from the right. Learn about one-sided limits, infinite limits, and how to determine the behavior of rational functions near vertical asymptotes. Suitable for students in Grades 11-12.

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