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The given problem asks for the evaluation of the following limit:

\[
\lim_{\theta \to 9\pi} \sqrt{\theta + 81 \cos(\theta + 9\pi)}
\]

### Step-by-step solution:

1. **Simplify the cosine term:**
   We know that \( \cos(\theta + 9\pi) = \cos(\theta) \) because \( \cos \) has a periodicity of \( 2\pi \), and \( 9\pi \) is an odd multiple of \( \pi \). Therefore:

   \[
   \cos(\theta + 9\pi) = \cos(\theta + \pi) = -\cos(\theta)
   \]
   So, we can rewrite the expression as:

   \[
   \lim_{\theta \to 9\pi} \sqrt{\theta + 81 \cos(\theta + 9\pi)} = \lim_{\theta \to 9\pi} \sqrt{\theta - 81 \cos(\theta)}
   \]

2. **Substitute \( \theta = 9\pi \):**
   Next, we substitute \( \theta = 9\pi \) into the simplified expression:

   \[
   \sqrt{9\pi + 81 \cos(9\pi + 9\pi)} = \sqrt{9\pi + 81 \cdot (-1)}
   \]

   \[
   = \sqrt{9\pi - 81}
   \]

3. **Approximate the result:**
   Now, we compute the value of the expression inside the square root. Since \( 9\pi \) is approximately 28.27:

   \[
   9\pi - 81 = 28.27 - 81 = -52.73
   \]

   Since the value inside the square root is negative, this means the expression is not defined in the real number system.

### Conclusion:
The limit does not exist (DNE) in the real number system because the square root of a negative number results in an imaginary number.

Would you like to see more details or have any further questions?

### Related questions:
1. What is the limit if we replace the cosine function with a sine function in a similar expression?
2. How does the concept of periodicity in trigonometric functions help simplify limit problems?
3. What happens when evaluating the limit if the inside of the square root turns out to be positive?
4. How do we handle limits that lead to imaginary numbers?
5. Could we extend this limit to the complex number system, and if so, what would the result be?

**Tip**: Always consider the domain restrictions, like square roots, when working with limits in real numbers.

Evaluate the following limit: lim(θ→9π) √(θ + 81 cos(θ + 9π))

This problem demonstrates how to evaluate the limit of the expression √(θ + 81 cos(θ + 9π)) as θ approaches 9π using trigonometric identities and properties of limits. Learn the step-by-step process, including simplifying trigonometric functions and handling negative values inside square roots.

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