Question

Consider the following statements:

I. $\lim_{n \to \infty} \frac{2^n + (-2)^n}{2^n}$ does not exist

II. $\lim_{n \to \infty} \frac{3^n + (-3)^n}{4^n}$ does not exist

Then:

• A. I is true and II is false
• B. I is false and II is true
• C. Both I and II are true
• D. Both I and II are false

Answer 1

Answer: (A)

I is true and II is false

Answer 2

To analyze the statements, we evaluate each limit separately.

Statement I: $\lim_{n \to \infty} \frac{2^n + (-2)^n}{2^n}$ does not exist.

Let's simplify the expression: $\frac{2^n + (-2)^n}{2^n} = \frac{2^n}{2^n} + \frac{(-2)^n}{2^n} = 1 + \left(\frac{-2}{2}\right)^n = 1 + (-1)^n$

Now, consider the behavior of $1 + (-1)^n$ as $n \to \infty$:

• If $n$ is an even integer (e.g., $n=2, 4, 6, \dots$), then $(-1)^n = 1$. So, $1 + (-1)^n = 1 + 1 = 2$.
• If $n$ is an odd integer (e.g., $n=1, 3, 5, \dots$), then $(-1)^n = -1$. So, $1 + (-1)^n = 1 - 1 = 0$.

Since the expression oscillates between two different values (2 and 0) as $n \to \infty$, the limit does not exist. Therefore, Statement I is true.

Statement II: $\lim_{n \to \infty} \frac{3^n + (-3)^n}{4^n}$ does not exist.

Let's analyze the expression $\frac{3^n + (-3)^n}{4^n}$.

• If $n$ is an even integer (let $n=2k$ for some integer $k \ge 1$): $\frac{3^{2k} + (-3)^{2k}}{4^{2k}} = \frac{3^{2k} + 3^{2k}}{4^{2k}} = \frac{2 \cdot 3^{2k}}{4^{2k}} = 2 \left(\frac{3^2}{4^2}\right)^k = 2 \left(\frac{9}{16}\right)^k$ As $n \to \infty$, $k \to \infty$. Since $\left|\frac{9}{16}\right| < 1$, we know that $\lim_{k \to \infty} \left(\frac{9}{16}\right)^k = 0$. Thus, $\lim_{n \to \infty, n \text{ even}} \frac{3^n + (-3)^n}{4^n} = 2 \cdot 0 = 0$.

• If $n$ is an odd integer (let $n=2k+1$ for some integer $k \ge 0$): $\frac{3^{2k+1} + (-3)^{2k+1}}{4^{2k+1}} = \frac{3^{2k+1} - 3^{2k+1}}{4^{2k+1}} = \frac{0}{4^{2k+1}} = 0$ As $n \to \infty$, the value of the expression is always 0. Thus, $\lim_{n \to \infty, n \text{ odd}} \frac{3^n + (-3)^n}{4^n} = 0$.

Since the limit approaches the same value (0) for both even and odd values of $n$, the limit $\lim_{n \to \infty} \frac{3^n + (-3)^n}{4^n}$ exists and is equal to 0. Therefore, Statement II, which claims the limit does not exist, is false.

Conclusion: Statement I is true. Statement II is false.