1.2:Rates of Change & Limits

Learning Goals:

©2009 Mark Pickering

•Calculate average & instantaneous speed•Define, calculate & apply properties of limits•Use Sandwich Theorem

Important Ideas•Limits are what make calculus different from algebra and trigonometry•Limits are fundamental to the study of calculus•Limits are related to rate of change•Rate of change is important in engineering & technology

Theorem 1Limits have the following properties:

lim ( )x c

g x M

lim ( )x c

f x L

if &then:

lim ( ) ( )x c

f x g x L M

1.

Theorem 1Limits have the following properties:

lim ( )x c

g x M

lim ( )x c

f x L

if &then:

lim ( ) ( )x c

f x g x L M

2.

Theorem 1Limits have the following properties:

lim ( )x c

g x M

lim ( )x c

f x L

if &then:

lim ( ) ( )x c

f x g x L M

3.

Theorem 1Limits have the following properties:

lim ( )x c

f x L

ifthen:

lim ( )x c

k f x k L

4.

& k a constant

Theorem 1Limits have the following properties:

lim ( )x c

g x M

lim ( )x c

f x L

if &then:

( )lim , 0( )x c

f x L Mg x M

5.

Theorem 1Limits have the following properties:

lim ( )x c

f x L

if &

6.

r & s areintegers, then:

lim ( )x c

rrssf x L

Theorem 1Limits have the following properties:if where k is a

7.constant, then:lim ( ) lim

x c x cf x k k

( )f x k

(not in your text as Th. 1)

Theorem 2For polynomial and rational functions:

lim ( ) ( )x c

f x f c

( ) ( )lim , ( ) 0( ) ( )x c

f x f c g cg x g c

a.b.

Limits may be found by substitution

ExampleSolve using limit properties and substitution:

2

3lim 2 3 2x

x x

Try ThisSolve using limit properties and substitution:2

2

4lim3x

x xx

6

ExampleSometimes limits do not exist. Consider:

3

2

3lim2x

xx

If substitution gives a constant divided by 0, the limit does not exist (DNE)

ExampleTrig functions may have limits.

2

lim(sin )x

x

Try This

2

lim(cos )x

x

2

lim(cos ) cos 02x

x

ExampleFind the limit if it exists:3

1

1lim

1x

xx

Try substitution

ExampleFind the limit if it exists:3

1

1lim

1x

xx

Substitution doesn’t work…does this mean the limit doesn’t exist?

Important Idea3 21 ( 1)( 1)

1 1x x x xx x

2 1x x and

are the same except at x=-1

Important Idea

The functions have the same limit as x-1

Procedure1.Try substitution2. Factor and cancel if

substitution doesn’t work

3.Try substitution againThe factor & cancellation

technique

Try ThisFind the limit if it exists:

2

3

6lim

3x

x xx

5Isn’t th

at

easy?Did you think calculus

was going to be

difficult?

Try ThisFind the limit if it exists:

22

2lim4x

xx

14

Try ThisFind the limit if it exists:

2

3

6lim

3x

x xx

The limit doesn’t existConfirm by graphing

DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.

Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.

Try ThisFind the limit if it exists:

2

1

2 3lim

1x

x xx

-5

Try ThisGraph and

3

111

xY

x

22 1Y x x on the

same axes. What is the difference between these graphs?

3 1( )

1x

f xx

Why is there a “hole” in the graph at x=1?

Analysis

ExampleConsider3 1

( )1

xf x

x

for ( ,1) (1, ) and

( ) 4f x

for x=13

1

1lim

1x

xx

=?

Try ThisFind: if

1lim ( )x

f x

2( ) 2, 1f x x x

( ) 1, 1f x x

1lim ( ) 3x

f x

Important IdeaThe existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.

Important IdeaWhat matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.

Try ThisFind:

f(0)is undefined; 2 is the limit

2( )

1 1x

f xx

0lim ( )x

f x

Find:

( ) 1, 0f x x

Try This( ) , 0

1 1x

f x xx

f(0) is defined; 2 is the limit

21

0lim ( )x

f x

Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:

2 , 3( )

3 , 3x

f xx

3lim ( ) 2x

f x

Try ThisGraph and find the limit (if it exists):

3

3lim

3x x DNE

Theorem 3: One-sided & Two Sided limits

if lim ( )x c

f x L

(limit from right)

andlim ( )x c

f x L

(limit from left)

then lim ( )x c

f x L

(overall limit)

Theorem 3: One-sided & Two Sided limits

(Converse)if lim ( )

x cf x L

(limit from

right)andlim ( )x c

f x M

(limit from left)

then lim ( )x c

f x

(DNE)

ExampleConsider

3 1( ) , 1

1x

f x xx

What happens at x=1?

x .75 .9 .99 .999

f(x)

Let x get close to 1 from the left:

Try ThisConsider

3 1( ) , 1

1x

f x xx

x 1.25 1.1 1.01

1.001

f(x)

Let x get close to 1 from the right:

Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?

3

1

1lim 3

1x

xx

Try ThisFind the limit if it exists:

0limx

xx

DNE

ExampleFind the limit if it exists:

0

1lim sinx x

Example1.Graph using a friendly window:

1sin

x

2. Zoom at x=03. Wassup at x=0?

Important Idea

If f(x) bounces from one value to another (oscillates) as x approaches c, the limit of f(x) does not exist at c:

Theorem 4: Sandwich (Squeeze) Theorem

Let f(x) be between g(x) & h(x) in an interval containing c. Iflim ( ) lim ( )

x c x cg x h x L

lim ( )x c

f x L

then:f(x) is “squeezed” to L

ExampleFind the limit if it exists:

0

sinlim

Where is in radians and in the interval,2 2

ExampleFind the limit if it exists:

0

sinlim

Substitution gives the indeterminate form…

ExampleFind the limit if it exists:

0

sinlim

Factor and cancel doesn’t work…

ExampleFind the limit if it exists:

0

sinlim

Maybe…the squeeze theorem…

Exampleg()=1

h()=cos

sin( )f

Example

0lim1 1

0

lim cos 1

&therefore…

0

sinlim 1

Two Special Trig Limits

0

sinlim 1

0

1 coslim 0

Memoriz

e

ExampleFind the limit if it exists:

0

tanlimx

xx

0 0

sin 1lim lim 1 1 1

cosx x

xx x

ExampleFind the limit if it exists:

0

sin(5 )limx

xx

0 0

sin(5 ) sin(5 )lim 5 5 lim 5 1 5

5 5x x

x xx x

Try ThisFind the limit if it exists:

0

3 3 coslimx

xx

0

Lesson CloseName 3 ways a limit may fail to exist.

Practice1. Sec 1.2 #1, 3, 8, 9-18, 28-38E (just find limit L), 39-42gc (graphing calculator), 43-45