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!