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) =