(b) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=8 ?$

(c) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=\infty$ ?

Step-by-Step Solution

(a)

To determine

To define: The term sequence.

Answer

A sequence is a collection of objects that are arranged in an order.

Explanation

Examples:

All even numbers forms a sequence such as
2,4,6,8,10,⋯.

All odd numbers forms a sequence such as
1,3,5,7,9,⋯.

The number of elements in the sequence is called “the length of the sequence” and the position of elements in a sequence is called “rank or index” of the sequence.

(b)

To determine

To describe: The meaning
limn→∞an=8.

Answer

The sequence
{an} is closer to 8 as n approaches the larger number
(∞).

Explanation

The sequence
{an} converges to 8 as n tends to infinity
(∞). Here, the limit of the sequence
{an} is 8.

(c)

To determine

To describe: The meaning
limn→∞an=∞.

Answer

The sequence
{an} becomes larger
(∞) as n becomes large
(∞).

Explanation

The sequence
{an} is diverges as it converges to
∞. That is, the limit of the sequence
{an} is infinity
(∞).