Look at website on slide 5 for review on deriving area of a circle formula

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• Look at website on slide 5 for review on deriving
  area of a circle formula
• Mean girls clip the limit does not exist
• https//www.youtube.com/watch?voDAKKQuBtDo

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Introduction to LimitsSection 12.1Youll need
a graphing calculator
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What is a limit? Lets discuss the derivation of
the area of a circle (and circumference)
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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.
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• http//www.mathopenref.com/circleareaderive.html

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• 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

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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!
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FYI

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

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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

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Numerical Examples
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Numerical Example 1

• Lets look at a sequence whose nth term is given
  by
• What will the sequence look like?
• ½ , 2/3, ¾, 4/5, .99/100,...99999/100000

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What is happening to the terms of the sequence?
½ , 2/3, ¾, 4/5, .99/100,.99999/100000
Will they ever get to 1?
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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?
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Graphical Examples
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Graphical Example 1
As x gets really, really big, what is happening
to the height, f(x)?
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As x gets really, really small, what is
happening to the height, f(x)?
Does the height, or f(x) ever get to 0?
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Graphical Example 2
As x gets really, really close to 2, what is
happening to the height, f(x)?
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Graphical Example 3
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-7
-4
Find
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Graphical Example 4
Use your graphing calculator to graph the
following
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Graphical Example 4
Find
TRACE what is it approaching? TABLE Set table
to start at 1.997 with increments of .001 (TBLSET)
As x gets closer and closer to 2, what is the
value of f(x) getting closer to?
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Does the value of f(x) exist when x 2?
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ZOOM Decimal
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Limits that Fail to Exist
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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.
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Nonexistence Example 2 Unbounded Behavior

• Discuss the existence of the limit

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Nonexistence Example 3 Oscillating Behavior

• Discuss the existence of the limit
• Put this into your calc
• set table to start at -.003 with increments of
  .001

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
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Common Types of Behavior Associated with
Nonexistence of a Limit
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When can I use substitution to find the limit?

• When you have a polynomial or rational function
  with nonzero denominators

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H Dub

• 12.1 3-22, 23-47odd