Warm Up Classify each polygon. 1. a polygon with three congruent sides
Introduction to Limits. What is a limit? A Geometric Example Look at a polygon inscribed in a circle...
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Transcript of Introduction to Limits. What is a limit? A Geometric Example Look at a polygon inscribed in a circle...
Introduction to LimitsIntroduction to Limits
What is a What is a limit?limit?
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.
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
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!
FYIFYI
Archimedes used this method WAY WAY before calculus to find
the area of a circle.
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
Numerical Numerical ExamplesExamples
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
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…
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
01
limn n
Graphical Graphical ExamplesExamples
Graphical Example 1Graphical Example 1
1( )f x
x
As x gets really, really big, what is happening to the height, f(x)?
As x gets really, really small, what is happening to the height, f(x)?
Does the height, or f(x) ever get to 0?
01
limx x
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
Find
7lim ( )x
f x
Graphical Example 3
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?
Does the function
exist when x = 2?
ln ln 2( )
2
xf x
x
2lim ( )x
f x
2lim ( ) 0.5x
f x
ZOOM DecimalZOOM Decimal
Limits that Limits that Fail to Fail to ExistExist
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
7lim ( )x
f x
Nonexistence Example 2
Discuss the existence of the limit
Nonexistence Example 3: Nonexistence Example 3: Unbounded BehaviorUnbounded Behavior
Discuss the existence of the limit
20
1limx x
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
Common Types of Behavior Common Types of Behavior Associated with Associated with Nonexistence of a LimitNonexistence of a Limit
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:
)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) =
)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) =