We have seen a common structure in several mathematical situations - the idea of a limit.

\(\blacktriangleright \) Decimal numbers or decimal expansions are a sequence of digits that go on forever. The sequence gets closer to a number. If you stop at any place, then you don’t have the limiting number.

\(\blacktriangleright \) Rational functions and exponential functions have an end-behavior, which appears as a horizontal asymptote on the graph. The function values get closer and closer to the asymptotic value. If you stop at any place, then you don’t have the limiting value.

\(\blacktriangleright \) Sequences do the same thing.

We can create a sequence of numbers. The numbers might get closer and closer to a limiting number. They might not.

In some cases, the sequence does not head toward a limiting number. In some cases, we might know the limiting number is there, but have little information about it. So, we end up with two questions.

• Does the sequence have a limiting value?
• If so, what is the number?

Learning Outcomes

In this section, students will

• recognize sequences can be generated by functions.
• compute limits of sequences.
• understand growth rates of basic sequences.
• use important terminology for sequences.
• apply the monotone convergence theorem.