Consider the following statements :
`I. lim_(n to oo) ( 2^n +(-2)^n)/(2^n) ` dos not exist
`II. lim_(n to oo) ( 3^n +(-3)^n)/(2^n) ` does not exist then

Options

A. I is true and II is fals
B. I is false and II is true
C. I and II are true
D. neither I nor II is true

Answer

To solve the given problem, we need to analyze the two statements regarding the limits as \( n \) approaches infinity.

### Step 1: Analyze the First Statement
We need to evaluate the limit:
\[ \lim_{n \to \infty} \frac{2^n + (-2)^n}{2^n} \]

**Solution:**
1. Rewrite the expression:
\[ \lim_{n \to \infty} \frac{2^n + (-2)^n}{2^n} = \lim_{n \to \infty} \left(1 + \frac{(-2)^n}{2^n}\right) \]
2. Simplify \(\frac{(-2)^n}{2^n}\):
\[ \frac{(-2)^n}{2^n} = (-1)^n \]
3. Therefore, the limit becomes:
\[ \lim_{n \to \infty} \left(1 + (-1)^n\right) \]
4. As \( n \) approaches infinity, \( (-1)^n \) oscillates between -1 and 1. Hence, the limit does not exist.

**Conclusion for Statement I:**
The first statement is **true**.

### Step 2: Analyze the Second Statement
Now we evaluate the limit:
\[ \lim_{n \to \infty} \frac{3^n + (-3)^n}{2^n} \]

**Solution:**
1. Rewrite the expression:
\[ \lim_{n \to \infty} \frac{3^n + (-3)^n}{2^n} = \lim_{n \to \infty} \left(\frac{3^n}{2^n} + \frac{(-3)^n}{2^n}\right) \]
2. Simplify each term:
\[ \frac{3^n}{2^n} = \left(\frac{3}{2}\right)^n \quad \text{and} \quad \frac{(-3)^n}{2^n} = (-1)^n \left(\frac{3}{2}\right)^n \]
3. Therefore, the limit becomes:
\[ \lim_{n \to \infty} \left(\left(\frac{3}{2}\right)^n + (-1)^n \left(\frac{3}{2}\right)^n\right) = \lim_{n \to \infty} \left(1 + (-1)^n\right) \left(\frac{3}{2}\right)^n \]
4. Since \(\left(\frac{3}{2}\right)^n\) approaches infinity as \( n \) approaches infinity, the limit diverges.

**Conclusion for Statement II:**
The second statement is **false** because the limit exists and approaches infinity.

### Final Conclusion
- Statement I is true.
- Statement II is false.

### Answer
The correct option is: **First statement is true and second is false.**