Introduction to Limits

Section 1.2

What is a What is a limit?limit?

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.

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!

FYI

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

circle.

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

Numerical Numerical ExamplesExamples

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, 5/6, ….99/100, 99999/100000…

1nn

What is happening to the terms of the sequence?

Will they ever get to 1?

lim11

n

nn

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

01limn n

Graphical Graphical ExamplesExamples

Graphical Example 1

1( )f xx

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?

01limx x

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

Find7

lim ( )x

f x

Graphical Example 3

ln ln 2( )2

xf xx

Use your graphing calculator to graph the following:

Graphical Example 3

2lim ( )x

f x

Find

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 xx

2lim ( )x

f x

2lim ( ) 0.5x

f x

ZOOM DecimalZOOM Decimal

Limits that Limits that Fail to ExistFail to Exist

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: Unbounded Behavior

Discuss the existence of the limit

20

1limx x

Nonexistence Example 4: Oscillating 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 Associated with Nonexistence of a

Limit