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The given limit is:

\[
\lim_{x \to 4} \frac{-1}{4 - x}
\]

### Solution:

As \( x \) approaches 4 from either side, the denominator \( (4 - x) \) approaches 0. Let's analyze the behavior from both sides:

1. **As \( x \to 4^+ \) (from the right):**
   - When \( x \) is slightly greater than 4, \( 4 - x \) is a small negative number.
   - Therefore, \( \frac{-1}{4 - x} \) becomes a large positive number because you are dividing a negative number by a small negative value.

2. **As \( x \to 4^- \) (from the left):**
   - When \( x \) is slightly less than 4, \( 4 - x \) is a small positive number.
   - Therefore, \( \frac{-1}{4 - x} \) becomes a large negative number because you are dividing a negative number by a small positive value.

### Conclusion:
- As \( x \to 4^+ \), the limit tends to \( +\infty \).
- As \( x \to 4^- \), the limit tends to \( -\infty \).

Thus, the limit **does not exist** because the left-hand and right-hand limits approach different infinities.

However, if you're considering the behavior from one specific side:
- If you meant \( x \to 4^+ \) (from the right), the limit is \( +\infty \).
- If you meant \( x \to 4^- \) (from the left), the limit is \( -\infty \).

Would you like more details or explanations on any part of this?

Here are 5 related questions to expand:
1. What is the formal definition of a limit at infinity?
2. How do we determine left-hand and right-hand limits?
3. What happens if both left-hand and right-hand limits go to the same infinity?
4. How does L'Hôpital's Rule apply to limits with indeterminate forms?
5. Can limits at infinity be finite?

**Tip:** When evaluating limits, always check for different behaviors on either side of the point, especially when the denominator tends to zero.

Analyze the behavior of the limit \frac{-1}{4 - x} as x approaches 4 from both sides. Understand why the limit does not exist, with insights into left-hand and right-hand limits, leading to infinity and negative infinity. Suitable for advanced high school math students (Grades 11-12).

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