2. Find for different values of a. lim n -> ∞ n(a + cos n*pi)

We are tasked with finding the limit:

\[
\lim_{n \to \infty} n(a + \cos(n\pi))
\]

### Step-by-Step Solution:

1. **Understanding the expression**: The cosine function \( \cos(n\pi) \) alternates between two values depending on whether \( n \) is even or odd:
   - If \( n \) is even, \( \cos(n\pi) = 1 \).
   - If \( n \) is odd, \( \cos(n\pi) = -1 \).

Thus, \( \cos(n\pi) \) alternates between 1 and -1 as \( n \) increases.

2. **Expression involving \( a \) and \( \cos(n\pi) \)**: 
   \[
   n(a + \cos(n\pi))
   \]
   This alternates between two expressions as \( n \) becomes large:
   - When \( n \) is even: \( n(a + 1) \),
   - When \( n \) is odd: \( n(a - 1) \).

3. **Behavior as \( n \to \infty \)**:
   - For even \( n \), the expression becomes \( n(a + 1) \), which grows without bound (approaches \( \infty \)) if \( a + 1 \neq 0 \). 
     - If \( a = -1 \), then the limit for even \( n \) is 0.
   - For odd \( n \), the expression becomes \( n(a - 1) \), which also grows without bound (approaches \( \infty \)) if \( a - 1 \neq 0 \).
     - If \( a = 1 \), the limit for odd \( n \) is 0.

### Conclusion:

- If \( a = 1 \), then:
  \[
  \lim_{n \to \infty} n(a + \cos(n\pi)) = 0 \quad \text{(for odd \( n \))}.
  \]
  But for even \( n \), it approaches \( \infty \).
  
- If \( a = -1 \), then:
  \[
  \lim_{n \to \infty} n(a + \cos(n\pi)) = 0 \quad \text{(for even \( n \))}.
  \]
  But for odd \( n \), it approaches \( \infty \).

- For any other values of \( a \), the expression oscillates between \( n(a + 1) \) and \( n(a - 1) \), both of which tend to infinity as \( n \to \infty \).

Therefore, **the limit only exists and equals 0 when \( a = 1 \) or \( a = -1 \)**, but in those cases, it depends on the parity of \( n \).

Would you like further clarification or a deeper explanation?

Here are 5 questions for further exploration:
1. How do alternating sequences behave in limits?
2. Can you describe how the cosine function behaves with increasing multiples of \( \pi \)?
3. What happens to oscillating functions as \( n \to \infty \)?
4. How do limits behave when a function grows unbounded?
5. How would this limit change for different periodic functions instead of \( \cos(n\pi) \)?

**Tip**: Always verify the behavior of periodic components when dealing with trigonometric functions in limits.

Explore the detailed step-by-step solution to finding the limit lim n -> ∞ n(a + cos(nπ)). This problem covers key concepts in limits, sequences, and trigonometry, and discusses how the cosine function alternates for different values of n. Suitable for calculus-level students.

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