To find the limit of the sequence $c(n) = a(n) \times b(n)$, we analyze the limits of $a(n)$ and $b(n)$ individually.

Case 1: Both Sequences Converge

If $\lim\limits_{n \to \infty} a(n) = A$ and $\lim\limits_{n \to \infty} b(n) = B$, then:
$\lim\limits_{n \to \infty} c(n) = \lim\limits_{n \to \infty} (a(n) \times b(n)) = A \times B.$

Case 2: One Sequence Diverges

• If $a(n)$ diverges to $\pm\infty$ and $b(n)$ converges to a nonzero limit $B$, then $c(n)$ also diverges to $\pm\infty$.
• If $a(n) \to 0$ and $b(n)$ is bounded, then $c(n) \to 0$.
• If both $a(n)$ and $b(n)$ diverge to infinity, then $c(n) \to \infty$ unless their growth rates balance in such a way that they converge to a finite limit.

Case 3: Indeterminate Forms

If one sequence approaches 0 while the other approaches infinity (e.g., $a(n) \to 0$ and $b(n) \to \infty$), then the limit depends on the rate of decay/growth of each sequence. This may require L'Hôpital’s Rule or asymptotic analysis.

Would you like a specific example solved?

Here are some related questions:

1. What happens if $a(n)$ and $b(n)$ oscillate?
2. How do we handle cases where limits do not exist?
3. Can we apply the squeeze theorem to determine the limit?
4. What if $a(n)$ and $b(n)$ are defined recursively?
5. How do logarithmic or exponential sequences affect the limit?

Tip: When facing an indeterminate form, consider rewriting the expression using logarithms, exponentials, or L'Hôpital’s Rule.