Chapter 1: Limits

Section 1.1:Limit of a Sequence An infinite sequence is the range of a function which has the set of natural numbers as its domain. n F(n)

Your turn….

Notes:

Convergent vs. Divergent sequences If the terms of an infinite sequence approach a unique finite value, that sequence is called a convergent sequence. A sequence which does not converge is called divergent.

Example 4

A ball is dropped from a height of 2m above flat ground. Each time it hits the ground, after falling a distance h, it rebounds 0.75h. The heights to which the ball rises form a sequence. What is the sequence and what is the limit.

Example 1: Find the limit

Example 2:

Notes

Example 3:

Your turn:

Section 1.2: Limit of a Series Recall from 30-1 that a series is the SUM of the terms of a sequence. An infinite series in the sum of the terms of an infinite sequence. In this section, we will be working with infinite geometric series. Infinite Geometric Sequence: (what does it look like) Infinite Geometric Series: Two different formulas Standard: (for n terms)If -1 < r <1 :

Example 1:

Your turn:

Converting recurring decimals into fractions: Question: Convert ……. into fractions …..= A=r= S=

Section 1.3: The Limit of a Function

Example 1: X F(x) X F(x)

Left- and Right- Hand Limits

Example 1 A parking lot charges $2.00 for the first hour and $1.00 for every subsequent hour. Draw a graph of the charges C (in dollars) versus time t (in hours). Does the limit as t->2 h exist? Explain. This graph is an example of a step function. This is a function which is constant throughout the intervals. What happens when you approach 2 h from the left? Important:

Example 2

Example 3

Example 4: Piece-wise function

Continuity:

Continuity

Note:

Example 2:

Example 3:

Limit Theorems

Example 1:

Example 2:

Limits of Algebraic functions

Limits of Infinity (asymptotes)