Introduction to Limits

1
Introduction to Limits
2
What is a limit?
3
A Geometric Example

• Look at a polygon inscribed in a circle

As the number of sides of the polygon increases,
the polygon is getting closer to becoming a
circle.
4

• If we refer to the polygon as an n-gon,
• where n is the number of sides we can make some
  mathematical statements
• As n gets larger, the n-gon gets closer to being
  a circle
• As n approaches infinity, the n-gon approaches
  the circle
• The limit of the n-gon, as n goes to infinity is
  the circle

5
The symbolic statement is
The n-gon never really gets to be the circle, but
it gets close - really, really close, and for all
practical purposes, it may as well be the circle.
That is what limits are all about!
6
FYI

• Archimedes used this method WAY before calculus
  to find the area of a circle.

7
An Informal Description

• If f(x) becomes arbitrarily close to a single
  number L as x approaches c from either side, the
  limit for f(x) as x approaches c, is L. This
  limit is written as

8
Numerical Examples
9
Numerical Example 1

• Lets look at a sequence whose nth term is given
  by
• What will the sequence look like?
• ½ , 2/3, ¾, 5/6, .99/100, 99999/100000

10
What is happening to the terms of the sequence?
½ , 2/3, ¾, 5/6, .99/100, 99999/100000
Will they ever get to 1?
11
Numerical Example 2
Lets look at the sequence whose nth term is
given by
1, ½, 1/3, ¼, ..1/10000, 1/10000000000000
As n is getting bigger, what are these terms
approaching?
12
(No Transcript)
13
Graphical Examples
14
Graphical Example 1
As x gets really, really big, what is happening
to the height, f(x)?
15
As x gets really, really small, what is
happening to the height, f(x)?
Does the height, or f(x) ever get to 0?
16
(No Transcript)
17
Graphical Example 2
As x gets really, really close to 2, what is
happening to the height, f(x)?
18
Graphical Example 3
Find
19

Graphical Example 3
Use your graphing calculator to graph the
following
Find
As x gets closer and closer to 2, what is the
value of f(x) getting closer to?
20
Does the function exist when x 2?
21
ZOOM Decimal
22
Limits that Fail to Exist
23
Nonexistence Example 1 Behavior that Differs
from the Right and Left
What happens as x approaches zero?
The limit as x approaches zero does not exist.
24
Nonexistence Example 2
Discuss the existence of the limit
25
Nonexistence Example 3 Unbounded Behavior

• Discuss the existence of the limit

26
Nonexistence Example 4 Oscillating Behavior

• Discuss the existence of the limit

X 2/p 2/3p 2/5p 2/7p 2/9p 2/11p X 0
Sin(1/x) 1 -1 1 -1 1 -1 Limit does not exist
27
Common Types of Behavior Associated with
Nonexistence of a Limit
28
Definition of Limit
If limx?cf(x) limx?c-f(x) L then,
limx?cf(x)L (Again, L must be a fixed, finite
number.)
Examples
29
(No Transcript)
30
(No Transcript)