2.1 day 1: Step Functions

“Miraculous Staircase”Loretto Chapel, Santa Fe, NM

Two 360o turns without support!

Limit notation: limx cf x L

“The limit of f of x as x approaches c is L.”

So:0

sinlim 1x

x

x

The limit of a function refers to the value that the function approaches, not the actual value (if any).

2

lim 2x

f x

not 1

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.

(See page 58 for details.)

For a limit to exist, the function must approach the same value from both sides.

One-sided limits approach from either the left or right side only.

1 2 3 4

1

2

At x=1: 1

lim 0x

f x

1

lim 1x

f x

1 1f

left hand limit

right hand limit

value of the function

1

limxf x

does not exist because the left and right hand limits do not match!

At x=2: 2

lim 1x

f x

2

lim 1x

f x

2 2f

left hand limit

right hand limit

value of the function

2

lim 1xf x

because the left and right hand limits match.

1 2 3 4

1

2

At x=3: 3

lim 2x

f x

3

lim 2x

f x

3 2f

left hand limit

right hand limit

value of the function

3

lim 2xf x

because the left and right hand limits match.

1 2 3 4

1

2

“Step functions” are sometimes used to describe real-life situations.

Our book refers to one such function: int( )y x

This is the Greatest Integer Function.

The TI-83 contains the command , but it is important that you understand the function rather than just entering it in your calculator.

int( )x

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

1.5 12 2

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

1.5 12 2

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

1.5 12 2

Greatest Integer Function:

greatest integer that is xy

The greatest integer function is also called the floor function.

The notation for the floor function is:

y x

Some books use or . y x y x

The calculator “connects the dots” which covers up the discontinuities.

The TI-83 command for the floor function is int (x).

Graph the floor function for and .8 8x 4 4y

Y=

CATALOG I int(

int (x)

Go to Y=

Highlight the slant line to the left.

ENTER

GRAPH

The open and closed circles do not show, but we can see the discontinuities.

The TI-83 command for the floor function is int (x).

Graph the floor function for and .8 8x 4 4y

Press until the line appears dotted

Least Integer Function:

least integer that is xy

x y

0 00.5 1

0.75 11 1

x y

0 00.5 1

0.75 11 1

1.5 22 2

Least Integer Function:

least integer that is xy

x y

0 00.5 1

0.75 11 1

1.5 22 2

Least Integer Function:

least integer that is xy

x y

0 00.5 1

0.75 11 1

1.5 22 2

Least Integer Function:

least integer that is xy

The least integer function is also called the ceiling function.

The notation for the ceiling function is:

y x

Least Integer Function:

least integer that is xy

The TI-89 command for the ceiling function is ceiling (x).

Don’t worry, there are not wall functions, front door functions, fireplace functions!