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The problem asks to find:

\[
\lim_{{x \to 5}} \left( (-x - 1)^4 - 5 \right)
\]

### Step-by-Step Solution:

1. **Substitute** \( x = 5 \) directly into the expression:

\[
(-5 - 1)^4 - 5
\]

2. **Simplify the expression inside the parentheses**:

\[
(-6)^4 - 5
\]

3. **Calculate the fourth power of \(-6\)**:

\[
(-6)^4 = 1296
\]

4. **Now subtract 5**:

\[
1296 - 5 = 1291
\]

### Final Answer:

\[
\lim_{{x \to 5}} \left( (-x - 1)^4 - 5 \right) = 1291
\]

Would you like further details or have any other questions? Here are some related questions for practice:

1. What happens if you substitute \(x = -1\) in the same limit expression?
2. How would the limit change if the exponent were odd, e.g., \((-x - 1)^3\)?
3. Can you compute the derivative of the function \(f(x) = (-x - 1)^4 - 5\)?
4. What is the limit as \(x \to 5\) for the function \( (-x - 1)^4\) alone?
5. How would the result differ if the function were \( (x + 1)^4 - 5\)?

**Tip**: Always try direct substitution for limits first, especially when the function is continuous at the point of interest!

Find lim as x approaches 5 of ((-x - 1)^4 - 5). (If the limit does not exist, enter ∅.)

Learn how to find the limit of the expression ((-x - 1)^4 - 5) as x approaches 5 using direct substitution. This problem demonstrates concepts of polynomial limits and continuity, suitable for high school calculus students.

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