Question

If $\{x_n\}_{n \ge 1}$ is a sequence of real numbers such that $\lim_{n \to \infty} \frac{x_n}{n} = 0.001$, then
 

• A. $\{x_n\}_{n \ge 1}$ is a bounded sequence
• B. $\{x_n\}_{n \ge 1}$ is an unbounded sequence
• C. $\{x_n\}_{n \ge 1}$ is a convergent sequence
• D. $\{x_n\}_{n \ge 1}$ is a monotonically decreasing sequence

Hint:

If $\lim_{n \to \infty} \frac{x_n}{n} = c \neq 0$, then $x_n$ grows approximately like $cn$ and hence is unbounded.

Answer 1

The Correct Option is B

Step 1: Given information.
We have $\lim_{n \to \infty} \frac{x_n}{n} = 0.001$. This implies that $\frac{x_n}{n}$ approaches $0.001$ as $n$ tends to infinity.

Step 2: Multiplying by $n$.
So, $x_n \approx 0.001n$ for large $n$. As $n \to \infty$, $x_n$ also tends to infinity because it grows linearly with $n$.

Step 3: Conclusion.
Since $x_n$ grows without bound, the sequence $\{x_n\}$ is unbounded.