introduction to limits. what is a limit? a geometric example look at a polygon inscribed in a circle...
TRANSCRIPT
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Introduction to LimitsIntroduction to Limits
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What is a What is a limit?limit?
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A Geometric ExampleA 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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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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FYIFYI
Archimedes used this method WAY WAY before calculus to find
the area of a circle.
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An Informal DescriptionAn 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 1Numerical Example 1
Let’s 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…
1
n
n
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What is happening to the What is happening to the terms of the sequence?terms of the sequence?
lim11
n
n
n
Will they ever get to 1?
½ , 2/3, ¾, 5/6, ….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 1Graphical Example 1
1( )f x
x
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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01
limx x
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Graphical Example 2Graphical 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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Find
7lim ( )x
f x
Graphical Example 3
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ln ln 2( )
2
xf x
x
Use your graphing calculator to graph the following:
Graphical Example 3
2lim ( )x
f xFind
As x gets closer and closer to 2, what is the value of f(x) getting closer to?
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Does the function
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 Fail to ExistExist
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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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7lim ( )x
f x
Nonexistence Example 2
Discuss the existence of the limit
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Nonexistence Example 3: Nonexistence Example 3: Unbounded BehaviorUnbounded Behavior
Discuss the existence of the limit
20
1limx x
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Nonexistence Example 4: Nonexistence Example 4: Oscillating BehaviorOscillating Behavior
Discuss the existence of the limit
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
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Common Types of Behavior Common Types of Behavior Associated with Associated with Nonexistence of a LimitNonexistence of a Limit
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Definition of Limit:Definition of Limit:
If limxc+f(x) = limxc-f(x) = L then,
limxcf(x)=L (Again, L must be a fixed, finite
number.)
f(2) =
lim ( )x
f x
2lim ( )
xf x
4
lim ( )x
f x
2
lim ( )x
f x
2
lim ( )x f x
lim ( )x
f x
4
lim ( )x f x 4
f(4) = lim ( )x f x
Examples:
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)x(flim
0xlim ( )
xf x
4
)x(flim
0x
)x(flim0x
lim ( )x f x
lim ( )x
f x
4
lim ( )x f x 4
f(4) =
lim ( )x f x
f(0) =
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)x(flim
3x
)x(flim6x
)x(flim
3x
)x(flim3x
lim ( )x f x
)x(flim
6x
)x(flim 6x
f(6) =
lim ( )x f x
f(3) =