1 1 limits of a function

Tran Huong Lan... After this lesson, you will know • Definition of limits • How to estimate a limit • When a limit exists and does not exist • How to interpret limits • How to estima

1.1 Limit of a Function

Chapter 1

Dr Tran Huong Lan

LIMIT=TENDENCY

First example

2 2

2

4

x

 



2

2

2 4

x

x

 



1.9 1.99 1.99 2 2.001 2.01 2.1 0.746 0.749 0.7499 || 0.750 0.751 0.756

We create a table of values for x close to 2

0.75

After this lesson, you will know

• Definition of limits

• How to estimate a limit

• When a limit exists and does not exist

• How to interpret limits

• How to estimate limits by graphing

Informal definition of limit

Example

x

-0.1 -0.01 -0.001 0 0.001 0.01 0.1

0.995 0.99995 0.9999995 1 0.9999995 0.99995 0.995

We create a table of values for x close to 0

0

cos x

sin x -0.01 -0.001 -0.0001 0 0.0001 0.001 0.01

1

Note that the limit at 0 IS NOT the value of the function at 0

One-sided limits

• Right-hand limit

• Left-hand limit

lim ( ) ( )

x c

f x L x c



lim ( ) ( )

x c

f x L x c



3

lim ( ) 1

x

h x



3

lim ( ) 1

x

h x



3

lim ( ) does not exist.

x h x



One-sided limit theorm

The two-sided limit lim ( ) if and only if lim ( ) lim ( )

x c

f x L





 

Example: find limits from graphs

Estimate the following limits

0

lim ( ) 1

x f x

1

lim ( ) does not exist

x g x

 lim ( )1 2

x h x

  

Example: find limits from graphs

Estimate the following limits when

sin x

y

x

sin

y

x



0

x 

2

1

y x



0

sin lim 1

x

x x

1 lim does not exist

1 lim sin does not exist

x x

Infinite limits

Formal definition of a limit

Epsilon-delta definition of a limit

The interval always contains c

L is the LIMIT

Epsilon-delta definition of a limit

The interval always contains c

L is the LIMIT

Epsilon-delta definition of a limit

The interval does not contain c !

NOT