Page 92 - Calculus for the Clueless, Calc II

Example 4—

Find the limit of {a n} = {ln (n + 1) - ln n}.

The sequence converges to 0.

Example 5—

n
Does {(-1) } have a limit?
This sequence is -1, +1,-1, +1,-1, .... There is no limit because the sequence does not go to one number.

Definition

if, given an ε > 0, there exists an N > 0, such that if n > N, | a n-L | < ε.

Note

It is not important that you know the technical definition of a limit to understand the rest of the chapter. But ...
at this point of your mathematical career, you should start understanding the background. It probably will help
you later on. It would also be nice if you could see the beauty and the depth of this material—the beginnings of
calculus. It truly is a wonderful discovery.

Example 6—

Using ε. N, show .

provided 3/ε < n + 1 or 3/ε - 1 < n. We then choose N as the whole-number part of 3/ε - 1.

The following theorems are used often. They are proved in many books and will only be stated here.

Let ; k = constant, f continuous. Then

1.

2.

3.

4.

5.

6.

Example 7—