look at website on slide 5 for review on deriving area of a circle formula mean girls clip: the...
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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?v=oDAKKQuBtDo
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Introduction to Limits
Section 12.1
You’ll need a graphing calculator
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What is a limit?What is a limit?Let’s discuss the Let’s discuss the derivation of the derivation of the area of a circlearea of a circle
(and circumference)(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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lim( )n
n go circlen
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 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
lim ( )x cf x L
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Numerical Numerical ExamplesExamples
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Numerical Example 1
Let’s 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…
1
n
n
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What is happening to the terms of the sequence?
Will they ever get to 1?
lim11
n
n
n
½ , 2/3, ¾, 4/5, ….99/100,….99999/100000…
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Let’s 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?
1n
Numerical Example 2
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01
limn n
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Graphical Graphical ExamplesExamples
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Graphical Example 1
1( )f x
x
As x gets really, really big, what is happening to the height, f(x)?
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01
lim xx
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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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01
limx x
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Graphical Example 2
3( )f x x
As x gets really, really close to 2, what is happening to the height, f(x)?
3
2im 8lxx
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Find7
lim ( )x
f x
Graphical Example 3
-4
-7
6
!6
4)(lim7
not
xfx
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ln ln 2( )
2
xf x
x
Use your graphing calculator to graph the following:
Graphical Example 4
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ln ln 2
( )2
xf x
x
Graphical Example 4
2lim ( )x
f x
Find
As x gets closer and closer to 2, what is the value of f(x) getting closer to?
TRACE: what is it approaching?TABLE:Set table to start at 1.997 with increments of .001 (TBLSET)
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Does the value of f(x)
exist when x = 2?
ln ln 2( )
2
xf x
x
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2lim ( )x
f x
2lim ( ) 0.5x
f x
ZOOM DecimalZOOM Decimal
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Limits that Limits that Fail to ExistFail to Exist
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What happens as x What happens as x approaches zero?approaches zero?
The limit as x approaches zero does not exist.
0
1limx
does not e tx
xis
Nonexistence Example 1: Behavior that Differs from the Right and Left
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Nonexistence Example 2: Unbounded Behavior
Discuss the existence of the limit
20
1limx x
0
1limx
does not e tx
xis
20
1limx x
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Nonexistence Example 3: Oscillating Behavior
0
1limsinx x
X 2/π 2/3π 2/5π 2/7π 2/9π 2/11π X 0
Sin(1/x) 1 -1 1 -1 1 -1 Limit does not exist
Discuss the existence of the limit
Put this into your calc
set table to start at -.003 with increments of .001
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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