https://file.aiquickdraw.com/mathbot/file/172991480711064kld9cr.jpg

The expression in the image represents a limit problem of the form:

\[
\lim_{{x \to 4^+}} \frac{-x^2 - 5x + 6}{2x - 8}
\]

### Step-by-step Solution:

1. **Substitute \( x = 4 \)** in the denominator to determine if there is a division by zero:
   \[
   2(4) - 8 = 8 - 8 = 0
   \]
   This shows that the denominator approaches zero as \( x \) approaches \( 4 \). Let’s analyze the numerator:

   \[
   -4^2 - 5(4) + 6 = -16 - 20 + 6 = -30
   \]

   Thus, the numerator approaches \( -30 \) as \( x \) approaches \( 4 \).

2. **Identifying the behavior**:
   - The numerator is not zero at \( x = 4 \), while the denominator approaches zero, indicating that this is likely a vertical asymptote. Since we are taking the limit from the right-hand side (\( 4^+ \)), we need to analyze the sign of the denominator for values slightly greater than \( 4 \).

   If \( x \) is slightly greater than \( 4 \) (say \( 4.1 \)), then:
   \[
   2(4.1) - 8 = 0.2
   \]

   So the denominator is positive for \( x \) approaching \( 4 \) from the right. However, the numerator remains \( -30 \), which is negative. This combination (negative numerator and positive denominator approaching zero) implies that the limit from the right is going to \( -\infty \).

### Conclusion:
\[
\lim_{{x \to 4^+}} \frac{-x^2 - 5x + 6}{2x - 8} = -\infty
\]

Would you like more details on this or have any specific questions? Here are some related questions:

1. How do you determine limits that involve vertical asymptotes?
2. What are the differences between left-hand and right-hand limits?
3. Can limits approaching infinity or negative infinity be formally defined?
4. What are other cases where the denominator approaching zero leads to a finite limit?
5. How can polynomial expressions in limits be simplified using factoring?

**Tip:** When solving limits, always first check if direct substitution leads to an indeterminate form, and then proceed with algebraic manipulations or sign analysis as needed.

Find the limit as x approaches 4 from the right of (-x^2 - 5x + 6) / (2x - 8).

Learn how to evaluate the limit of the function (-x^2 - 5x + 6) / (2x - 8) as x approaches 4 from the right. This problem involves limit analysis, handling vertical asymptotes, and understanding the behavior of rational functions as they approach indeterminate forms.

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