Limits and Continuity

Discontinue means….

A graph that is continuous can be drawn without lifting your pencil.

The limit at any point will be defined at all points on a continuous graph

That means we are free to use our difference quotient to create an algebraic model for all secants

So far (intentionally), all of our graphs have been continuous…

There are situations that are best modeled with graphs that

are called “discontinuous”

Generally speaking, it does not really cause any problems for us…

The only thing we need to be careful of, however, is if P or Q are placed on the point of discontinuity….

There are enough situations that involve discontinuity, that we need

to add them to our study…

Each section of a discontinuous function may need a different equation

Draw the graph:

One classification of Discontinuous Functions are called

“Piecewise Functions”

Consider speeding…

If you get convicted of speeding, you pay a fine, and demerit points are recorded

on your license.

Offence (km / h)

0 - 14 015 – 29 3 30 – 44 445 and over 6

Demerit (points)

Let x be the number of KM over

Let y be the number of demerit points

Offence (km / h)

0 < x < 15 y = 015 < x < 30 y = 3 30 < x < 45 y = 445 < x y =6

Demerit (points)

Create the following graph:

X0 15 30 45

Y

123

4

567

Kilometers over the speed limit

Demerit

Points

Why can’t there be two dots filled in on top of

each other?

X0 15 30 45

Y

123

4

567

Kilometers over the speed limit

Demerit

Points

?

0 1 2 3 4

x

y

5

4

3

2

1

lim f(x) =

x 2-

1

lim f(x) =

x 2+

3

Therefore the function

is not continuous at x = 2

Algebraic limits

lim (5)

x -1

52 x

= 5

lim (3x4 – 5x)

x 2= 3(2)4 – 5(2)

= 38

lim

x -3=

=

5)3(2

1= dne

lim

x 12356

2

2

xxxx Always try to

factor first

lim

x 1 )1)(2()1)(5(

xxxx

)21()51(

=

=4

An indeterminate form:

lim

x 0 xx 24

Substitution will create division by zero, and we can not factor, so we will rationalize the numerator.

2424

xx

X

=41

)24(

44

xx

x

)24(

xx

x

24

1

x

Take the limit:

204

1

24

1

22

1

An indeterminate form:

Pg 44Read ex 2,3,4

4,5,8,9,[10-14] e.o.l.16